Convergence and continuity
Limit superior and limit inferior of a sequence (Tag D-HOKQD)
Let \((a_n)_{n\geq 0}\) be a sequence of real numbers. Its limit superior and limit inferior are \[ \begin{aligned} \limsup_{n\to\infty} a_n &\coloneqq \lim_{n\to \infty} \sup_{j\geq n} a_j = \inf_{n\geq 0} \sup_{j\geq n} a_j, \\ \liminf_{n\to\infty} a_n &\coloneqq \lim_{n\to \infty} \inf_{j\geq n} a_j = \sup_{n\geq 0} \inf_{j\geq n} a_j, \end{aligned} \] with values in \([-\infty,\infty]\); the limits exist and equal the infimum and supremum because \(\sup_{j\geq n}a_j\) is nonincreasing and \(\inf_{j\geq n}a_j\) is nondecreasing in \(n\).
Continuity and uniform continuity (Tag D-HHVPT)
Let \(X\subseteq\RR\) and let \(f\colon X\to \RR\).
The function \(f\) is continuous on \(X\) if \[ \forall x_0\in X \quad \forall \varepsilon>0 \quad \exists \delta=\delta(\varepsilon, x_0)>0 \quad \forall y\in X: \quad \abs{x_0 - y} < \delta \implies \abs{f(x_0) - f(y)} < \varepsilon. \]
The function \(f\) is uniformly continuous on \(X\) if \[ \forall \varepsilon>0 \quad \exists \delta=\delta(\varepsilon)>0 \quad \forall x, y \in X: \quad \abs{x - y} < \delta \implies \abs{f(x) - f(y)} < \varepsilon. \]
Substituting \(y = x_0 + h\), the two conditions read \[ \begin{aligned} &\forall x_0\in X \quad \forall \varepsilon>0 \quad \exists \delta(\varepsilon, x_0)>0 \quad \forall h \text{ with } x_0+h\in X: && \abs{h} < \delta \implies \abs{f(x_0) - f(x_0 + h)} < \varepsilon, \\ &\forall \varepsilon>0 \quad \exists \delta(\varepsilon)>0 \quad \forall x\in X \quad \forall h \text{ with } x+h\in X: && \abs{h} < \delta \implies \abs{f(x) - f(x + h)} < \varepsilon. \end{aligned} \] In the first, \(\delta\) may depend on the point \(x_0\); in the second, one \(\delta\) serves every point of \(X\).
Uniform continuity (Tag FD-ST7TD)
Let \((X, d_1)\) and \((Y, d_2)\) be metric spaces. A function \(f\colon X \to Y\) is uniformly continuous if for every \(\varepsilon > 0\) there exists \(\delta=\delta(\varepsilon) > 0\) such that for all \(x,y\in X\), \[ x\in B_\delta(y) \implies f(x) \in B_\varepsilon(f(y)). \]
For continuity of \(f\) at every point, the same condition is required with \(\delta=\delta(\varepsilon,y)\) allowed to depend on the point \(y\); uniform continuity requires one \(\delta\) for all \(y\in X\).
In the definition of continuity on a set, \(\delta\) may depend on both \(\varepsilon\) and the point \(x_0\); in the definition of uniform continuity, \(\delta\) depends only on \(\varepsilon\), and the condition holds for all pairs \(x,y\).
Topological notions in a metric space (Tag D-TVKFM)
Let \((X,d)\) be a metric space, let \(A\subseteq X\), and let \(p\in X\).
A neighborhood of \(p\) is an open set \(U_p\subseteq X\) containing \(p\).
An \(\varepsilon\)-neighborhood of \(p\) is an open ball \(B_\varepsilon(p) \coloneqq \theset{q\in X \suchthat d(p, q) < \varepsilon}\) for some \(\varepsilon>0\).
The point \(p\) is an accumulation point or a limit point of \(A\) if every punctured neighborhood \(U_p\setminus\theset{p}\) contains a point \(q\in A\). Write \(A'\) for the set of limit points of \(A\) and \(\overline{A} \coloneqq A\cup A'\) for the closure of \(A\).
If \(p\in A\) and \(p\) is not a limit point of \(A\), then \(p\) is an isolated point of \(A\).
\(A\) is closed if \(A' \subseteq A\), so that \(A\) contains all of its limit points.
A point \(p\in A\) is an interior point of \(A\) if there is a neighborhood \(U_p\) of \(p\) with \(U_p \subseteq A\).
\(A\) is open if every point of \(A\) is an interior point of \(A\).
\(A\) is perfect if \(A\) is closed and \(A\subseteq A'\), so that every point of \(A\) is a limit point of \(A\).
\(A\) is bounded if there are a real number \(M\) and a point \(q\in X\) such that \(d(p, q) < M\) for all \(p\in A\).
\(A\) is dense in \(X\) if every point of \(X\) lies in \(A\) or is a limit point of \(A\), that is, \(X = A\cup A' = \overline{A}\).
Diameter of a subset of a metric space (Tag FD-5I73B)
Let \((X,d)\) be a metric space and let \(A\subseteq X\) be nonempty. The diameter of \(A\) is \[ \diam(A) \coloneqq \sup_{x, y\in A} d(x, y) \in [0,\infty]. \]
Diameter of a set (Tag FF-7JOOD)
Let \((X, d)\) be a metric space and let \(A\subseteq X\) be nonempty. The diameter of \(A\) is \[ \diam(A) = \sup_{x, y\in A} d(x, y)\in[0,\infty]. \]
Pointwise convergence (Tag D-IYDZU)
Let \(S\) be a set, and let \(f_n\colon S\to\CC\) for \(n\geq 1\) and \(f\colon S\to\CC\) be functions. The sequence \((f_n)\) converges pointwise to \(f\) on \(S\) if \[ (\forall \varepsilon>0)\,(\forall x \in S)\,(\exists n_0 = n_0(x, \varepsilon))\,(\forall n>n_0)\quad \abs{f_n(x)-f(x)}<\varepsilon. \]
Uniform convergence (Tag D-YZC3C)
Let \(S\) be a set, and let \(f_n\colon S\to\CC\) for \(n\geq 1\) and \(f\colon S\to\CC\) be functions. The sequence \((f_n)\) converges uniformly to \(f\) on \(S\) if \[ (\forall \varepsilon>0)\,(\exists n_0 = n_0(\varepsilon))\,(\forall x \in S)\,(\forall n>n_0)\quad \abs{f_n(x)-f(x)}<\varepsilon. \]
Negating the quantifiers, \((f_n)\) does not converge uniformly to \(f\) on \(S\) if and only if \[ (\exists \varepsilon>0)\,(\forall n_0)\,(\exists x = x(n_0) \in S)\,(\exists n>n_0)\quad \abs{f_n(x)-f(x)} \geq \varepsilon. \] The point \(x\) at which the estimate fails may depend on \(n_0\).
Uniform convergence of a sequence of functions (Tag FD-ZTRHG)
Let \(E\) be a set, and let \(f_n\colon E\to\CC\) for \(n\geq 1\) and \(f\colon E\to\CC\) be functions. The sequence \((f_n)\) converges uniformly to \(f\) on \(E\) if for every \(\varepsilon>0\) there exists \(N\in\NN\) such that \(\abs{f_n(x) - f(x)} < \varepsilon\) for all \(n\geq N\) and all \(x\in E\).
Let \(E\), \((f_n)\), and \(f\) be as in the definition, and put \(\norm{f_n - f}_\infty \coloneqq \sup_{x\in E}\abs{f_n(x) - f(x)}\in[0,\infty]\). Then \((f_n)\) converges uniformly to \(f\) on \(E\) if and only if \(\norm{f_n - f}_\infty\to 0\) as \(n\to\infty\).
If \((f_n)\) converges uniformly to \(f\), then for \(\varepsilon>0\) and \(N\) as in the definition, \(\norm{f_n - f}_\infty \leq \varepsilon\) for all \(n\geq N\), so \(\norm{f_n - f}_\infty\to 0\). Conversely, if \(\norm{f_n - f}_\infty\to 0\), then for \(\varepsilon>0\) there exists \(N\) with \(\norm{f_n - f}_\infty < \varepsilon\) for all \(n\geq N\), and then \(\abs{f_n(x) - f(x)}\leq\norm{f_n - f}_\infty<\varepsilon\) for all \(n\geq N\) and \(x\in E\).
Uniform convergence of a sequence of functions (Tag FD-F3UU4)
Let \(E\) be a set, and let \(f_n\colon E\to\RR\) for \(n\geq 1\) and \(f\colon E\to\RR\) be functions. The sequence \((f_n)\) converges uniformly to \(f\) on \(E\) if for every \(\varepsilon>0\) there exists \(N=N(\varepsilon)\) such that \(\abs{f_n(x) - f(x)} < \varepsilon\) for all \(n\geq N\) and all \(x\in E\).
With \(\norm{g}_\infty \coloneqq \sup_{x\in E}\abs{g(x)}\) for \(g\colon E\to\RR\), the sequence \((f_n)\) converges uniformly to \(f\) on \(E\) if and only if \(\norm{f_n - f}_\infty\to 0\) as \(n\to\infty\).
Function spaces
Completeness of a metric space (Tag D-G5N6I)
A metric space \((X,d)\) is complete if every Cauchy sequence in \(X\) converges to a point of \(X\).
Completeness of a normed space via absolutely convergent series (Tag FD-WN55Z)
Let \((V,\norm{\cdot})\) be a normed vector space. The following are equivalent:
\(V\) is complete: every Cauchy sequence in \(V\) converges.
Every absolutely convergent series in \(V\) converges: if \(x_n\in V\) and \(\sum_{n=1}^\infty\norm{x_n}<\infty\), then \(\sum_{n=1}^\infty x_n\) converges in \(V\).
Assume \(V\) is complete and \(\sum_n\norm{x_n}<\infty\). For \(M>N\) the partial sums \(s_N\coloneqq\sum_{n=1}^N x_n\) satisfy \(\norm{s_M-s_N}\leq\sum_{n=N+1}^M\norm{x_n}\), so \((s_N)\) is Cauchy and converges.
Conversely, assume every absolutely convergent series converges, and let \((y_m)\) be a Cauchy sequence. Choose \(m_1<m_2<\cdots\) with \(\norm{y_p-y_q}<2^{-k}\) for all \(p,q\geq m_k\). The series \(y_{m_1}+\sum_{k\geq 1}(y_{m_{k+1}}-y_{m_k})\) is absolutely convergent, so it converges; its partial sums are \(y_{m_{k}}\), so the subsequence \((y_{m_k})\) converges to some \(y\in V\). A Cauchy sequence with a convergent subsequence converges to the same limit, so \(y_m\to y\).
Completeness of a normed space via absolutely convergent series (Tag FF-HORGP)
Let \((V,\norm{\cdot})\) be a normed vector space. Then \(V\) is complete if and only if every absolutely convergent series in \(V\) converges: whenever \(x_n\in V\) and \(\sum_{n=1}^\infty\norm{x_n}<\infty\), the series \(\sum_{n=1}^\infty x_n\) converges in \(V\).
This is the equivalence proved in FD-WN55Z.
If \(X\) is a Banach space and \(\sum_n\norm{x_n}<\infty\), then \(\sum_n x_n\) converges in \(X\).
Dense subsets (Tag FD-BA2WU)
Let \(X\) be a topological space. A subset \(A\subseteq X\) is dense in \(X\) if its closure is \(X\): \(\cl_X(A) = X\).
Dense subset (Tag FF-NRCZN)
Let \(X\) be a topological space. A subset \(A\subseteq X\) is dense in \(X\) if and only if \(\cl_X(A) = X\), if and only if every nonempty open subset of \(X\) meets \(A\).
Separable spaces (Tag FD-Q7EEG)
A topological space \(X\) is separable if \(X\) has a countable dense subset.
Nowhere dense sets (Tag D-2MJRE)
Let \(X\) be a topological space. A subset \(S\subseteq X\) is nowhere dense in \(X\) if its closure \(\overline{S}\) has empty interior, that is, no nonempty open subset of \(X\) is contained in \(\overline{S}\).
Equivalently, every nonempty open set \(U\subseteq X\) contains a nonempty open set \(V\) with \(V\cap S=\emptyset\), so \(S\) is not dense in any nonempty open subset of \(X\). Indeed, if \(\overline{S}\) has empty interior then \(V\coloneqq U\setminus\overline{S}\) is nonempty and open; conversely, a nonempty open \(W\subseteq\overline{S}\) contains no nonempty open set disjoint from \(S\). For \(X=\RR\), \(S\) is nowhere dense if and only if every open interval contains an open subinterval disjoint from \(S\).
Nowhere dense sets (Tag FD-SMWLB)
Let \(X\) be a topological space. A set \(A\subseteq X\) is nowhere dense in \(X\) if its closure has empty interior: \(\qty{\overline{A}}^\circ=\emptyset\).
Equivalently, \(A\) is not dense in any nonempty open subset of \(X\). For \(X=\RR\), a set \(A\) is nowhere dense if and only if every open interval \(I\) contains an open subinterval \(S\subseteq I\) with \(S\cap A = \emptyset\), that is, \(\overline{A}\) contains no open interval.
In \(\RR\), the sets \(\theset{1/n \suchthat n\geq 1}\) and \(\ZZ\) are nowhere dense: their closures \(\theset{0}\cup\theset{1/n\suchthat n\geq 1}\) and \(\ZZ\) contain no open interval. The sets \(\QQ\) and \(\ZZ\cup\qty{(a, b)\cap \QQ}\) with \(a<b\) are not nowhere dense: their closures contain \(\RR\) and \([a,b]\) respectively.
Nowhere dense sets (Tag FF-HRUA3)
Let \(X\) be a topological space and \(A\subseteq X\). The following are equivalent:
\(A\) is nowhere dense in \(X\), that is, \(\qty{\overline{A}}^\circ = \emptyset\);
every nonempty open set \(U\subseteq X\) contains a nonempty open set \(V\) with \(V\cap A = \emptyset\);
\(A\) is not dense in any nonempty open subset of \(X\).
For \(X = \RR\), \(A\) is nowhere dense if and only if every open interval \(I\) contains an open interval \(S\subseteq I\) with \(S\cap A = \emptyset\), that is, \(\overline{A}\) contains no open interval.
$\{1/n\}$ and $\ZZ$ are nowhere dense; $\QQ$ is not (Tag FF-YXTQY)
In \(\RR\):
\(\theset{1/n\suchthat n\geq 1}\) and \(\ZZ\) are nowhere dense, since their closures \(\theset{0}\cup\theset{1/n\suchthat n\geq 1}\) and \(\ZZ\) contain no open interval;
\(\QQ\) and \(\ZZ\cup\qty{(a, b)\cap\QQ}\) for \(a<b\) are not nowhere dense, since their closures \(\RR\) and \(\ZZ\cup[a,b]\) contain the open interval \((a,b)\).
Meager sets (Tag D-5NODS)
Let \(X\) be a topological space. A subset \(S\subseteq X\) is meager in \(X\) if it is a countable union of nowhere dense subsets of \(X\).
Meagre sets (Tag FD-QEQIY)
Let \(X\) be a topological space. A subset \(A\subseteq X\) is meagre in \(X\) if it is a countable union of nowhere dense subsets of \(X\).
Meagre sets (Tag FF-SZR6O)
Let \(X\) be a topological space and \(A\subseteq X\). Then \(A\) is meagre in \(X\) if and only if \(A\) is a countable union of nowhere dense subsets of \(X\).
First and second category (Tag FD-JGBSF)
Let \(X\) be a topological space and let \(A\subseteq X\).
\(A\) is of first category in \(X\) if it is a countable union of nowhere dense subsets of \(X\).
\(A\) is of second category in \(X\) if it is not of first category in \(X\).
First and second category sets (Tag FF-QLQYF)
Let \(X\) be a topological space and \(A\subseteq X\). Then \(A\) is of first category in \(X\) if and only if \(A\) is a countable union of nowhere dense subsets of \(X\), and of second category in \(X\) otherwise.
Baire space (Tag D-VFNTY)
A topological space \(X\) is a Baire space if every countable intersection of open dense subsets of \(X\) is dense in \(X\).
Baire space (Tag FD-WHWSQ)
A topological space \(X\) is a Baire space if whenever \((U_n)_{n\geq 1}\) is a countable family of open dense subsets of \(X\), the intersection \(\bigcap_{n\geq 1} U_n\) is dense in \(X\).
Baire spaces (Tag FF-VON2D)
A topological space \(X\) is a Baire space if and only if for every countable family \((U_n)_{n\geq 1}\) of open dense subsets of \(X\), the intersection \(\bigcap_{n\geq 1} U_n\) is dense in \(X\).
Measure theory
Limit superior and limit inferior of a sequence of sets (Tag D-PAEDW)
Let \((E_n)_{n\geq 1}\) be a sequence of subsets of a set \(X\). Its limit inferior and limit superior are \[ \begin{aligned} \liminf_{n\to\infty} E_{n} &\coloneqq \bigcup_{N=1}^\infty \bigcap_{n=N}^\infty E_{n} = \theset{x\in X \suchthat x\in E_{n} \text{ for all but finitely many } n}, \\ \limsup_{n\to\infty} E_{n} &\coloneqq \bigcap_{N=1}^\infty \bigcup_{n=N}^{\infty} E_{n} = \theset{x\in X \suchthat x\in E_{n} \text{ for infinitely many } n}. \end{aligned} \]
Limit superior and limit inferior of a sequence of sets (Tag FD-AI6XN)
Let \((A_n)_{n\geq 1}\) be a sequence of subsets of a set \(X\). Its limit superior and limit inferior are \[ \begin{aligned} \limsup_{n\to\infty} A_n &\coloneqq \bigcap_{n\geq 1} \bigcup_{j\geq n} A_j = \theset{x\in X \suchthat x\in A_n \text{ for infinitely many } n}, \\ \liminf_{n\to\infty} A_n &\coloneqq \bigcup_{n\geq 1} \bigcap_{j\geq n} A_j = \theset{x\in X \suchthat x\in A_n \text{ for all but finitely many } n}. \end{aligned} \]
$\limsup$ and $\liminf$ of a sequence of sets (Tag FF-JBCFQ)
Let \((A_n)_{n\geq 1}\) be a sequence of subsets of a set \(X\). Its limit inferior and limit superior are \[ \liminf _{n \to \infty} A_n=\bigcup_{n \geq 1} \bigcap_{j \geq n} A_j, \qquad \limsup _{n \to \infty} A_n=\bigcap_{n \geq 1} \bigcup_{j \geq n} A_j. \] A point \(x\in X\) lies in \(\liminf_n A_n\) if and only if \(x\in A_j\) for all but finitely many \(j\), and in \(\limsup_n A_n\) if and only if \(x\in A_j\) for infinitely many \(j\). In particular, \(\liminf _{n \to \infty} A_n \subseteq \limsup _{n \to \infty} A_n\).
A point lies in \(\bigcap_{j\geq n}A_j\) for some \(n\) exactly when it lies in every \(A_j\) with \(j\geq n\) for some \(n\), that is, in all but finitely many \(A_j\). A point lies in \(\bigcup_{j\geq n}A_j\) for every \(n\) exactly when for every \(n\) it lies in some \(A_j\) with \(j\geq n\), that is, in infinitely many \(A_j\). A point in all but finitely many \(A_j\) lies in infinitely many of them.
For sets \(E_n\subseteq X\), \[ \liminf_n E_n = \bigcup_{N\geq1}\bigcap_{n\geq N}E_n, \qquad \limsup_n E_n = \bigcap_{N\geq1}\bigcup_{n\geq N}E_n, \] so \(x\in\liminf_nE_n\) if and only if \(x\in E_n\) for all but finitely many \(n\), and \(x\in\limsup_nE_n\) if and only if \(x\in E_n\) for infinitely many \(n\). These parallel \(\liminf_n x_n = \sup_N\inf_{n\geq N}x_n\) and \(\limsup_n x_n = \inf_N\sup_{n\geq N}x_n\), with suprema corresponding to unions and infima to intersections. In particular \(\liminf_n E_n\subseteq\limsup_n E_n\).
For a measure \(\mu\) and measurable \(E_n\), \[ \mu\qty{\liminf_n E_n }\leq \liminf_n \mu(E_n), \] and if \(\mu\qty{\bigcup_nE_n}<\infty\), then also \(\limsup_n\mu(E_n)\leq\mu\qty{\limsup_n E_n}\). If \(\liminf_nE_n = \limsup_nE_n\eqqcolon E\) and \(\mu\qty{\bigcup_nE_n}<\infty\), then \(\mu(E) = \lim_n\mu(E_n)\). For \(E_n\nearrow E=\bigcup_nE_n\) the equality \(\mu(E)=\lim_n\mu(E_n)\) holds for every measure \(\mu\), and for \(E_n\searrow E=\bigcap_nE_n\) it holds when \(\mu(E_1)<\infty\).
$F_\sigma$ and $G_\delta$ sets (Tag D-RPOGQ)
Let \(X\) be a topological space. A subset of \(X\) is an \(F_\sigma\) set if it is a countable union of closed sets, and a \(G_\delta\) set if it is a countable intersection of open sets.
In \(F_\sigma\), “F” stands for fermé, French for “closed”, and \(\sigma\) stands for a sum, that is, a union.
$F_\sigma$ sets (Tag FD-BBR6Q)
Let \(X\) be a topological space. A subset \(A\subseteq X\) is an \(F_\sigma\) set if \(A\) is a countable union of closed subsets of \(X\).
$G_\delta$ sets (Tag FD-HGESN)
Let \(X\) be a topological space. A subset \(A\subseteq X\) is a \(G_\delta\) set if \(A\) is a countable intersection of open subsets of \(X\).
Borel sets (Tag FD-V5ISD)
Let \(X\) be a topological space. The Borel \(\sigma\)-algebra of \(X\) is the smallest \(\sigma\)-algebra of subsets of \(X\) that contains every open subset of \(X\), namely the intersection of all such \(\sigma\)-algebras. Its elements are the Borel sets of \(X\).
Since a \(\sigma\)-algebra is closed under complements, countable unions, and countable intersections, every closed set, every $F_\sigma$ set, and every $G_\delta$ set is a Borel set.
Lebesgue outer measure (Tag D-3XE77)
A closed cube in \(\RR^n\) is a product \(Q=\prod_{i=1}^n[a_i,a_i+\ell]\) with \(\ell\geq 0\), and its volume is \(\abs{Q}\coloneqq\ell^n\). The outer measure of a set \(E\subseteq\RR^n\) is \[ m_*(E) \coloneqq \inf \theset{\sum_{i=1}^\infty \abs{Q_i} \suchthat Q_1, Q_2, \ldots \text{ closed cubes with } E\subseteq\bigcup_{i=1}^\infty Q_i} \in [0,\infty]. \]
Lebesgue outer measure (Tag FD-QA5ME)
For a closed cube \(Q=\prod_{i=1}^n[a_i,a_i+\ell]\subseteq\RR^n\), write \(\abs{Q}\coloneqq\ell^n\) for its volume. The outer measure of \(E\subseteq\RR^n\) is \[ m_*(E) \coloneqq \inf \theset{ \sum_{i=1}^\infty \abs{Q_i} \suchthat Q_1,Q_2,\ldots \text{ closed cubes with } E\subseteq\bigcup_{i=1}^\infty Q_i}. \]
Basic properties of Lebesgue outer measure (Tag FD-KNEDG)
Let \(m_*\) be the outer measure on \(\RR^n\).
Monotonicity. If \(E_1\subseteq E_2\subseteq\RR^n\), then \(m_*(E_1)\leq m_*(E_2)\).
Countable subadditivity. If \(E=\bigcup_{j=1}^\infty E_j\) with \(E_j\subseteq\RR^n\), then \(m_*(E)\leq\sum_{j=1}^\infty m_*(E_j)\).
Approximation from above by open sets. For every \(E\subseteq\RR^n\), \(m_*(E)=\inf\theset{m_*(O)\suchthat O\supseteq E \text{ open}}\).
Additivity for almost disjoint cubes. If \(E=\bigcup_{j=1}^\infty Q_j\) is a countable union of closed cubes that are pairwise almost disjoint, then \(m_*(E)=\sum_{j=1}^\infty\abs{Q_j}\).
Lebesgue measurable sets (Tag D-MDJII)
Let \(m_*\) be the outer measure on \(\RR^n\). A subset \(E\subseteq \RR^n\) is Lebesgue measurable if for every \(\varepsilon> 0\) there exists an open set \(O \supseteq E\) such that \(m_*(O\setminus E) < \varepsilon\). For a Lebesgue measurable set \(E\), its Lebesgue measure is \(m(E) \coloneqq m_*(E)\).
Lebesgue measurability of a set (Tag FD-D4S7E)
Let \(m_*\) be the outer measure on \(\RR^n\). A set \(E\subseteq \RR^n\) is measurable if for every \(\varepsilon>0\) there exists an open set \(G \supseteq E\) with \(m_*(G\setminus E)<\varepsilon\).
Equivalent characterizations of measurability of a set (Tag FT-KXFMK)
Let \(m_*\) be the outer measure on \(\RR^n\) and let \(E\subseteq \RR^n\). The following are equivalent:
\(E\) is Lebesgue measurable: for every \(\varepsilon>0\) there exists an open set \(G\supseteq E\) with \(m_*(G\setminus E)<\varepsilon\).
For every \(\varepsilon>0\) there exists a closed set \(F\subseteq E\) with \(m_*(E\setminus F) < \varepsilon\).
\(E = H \cup Z\) with \(H\) an $F_\sigma$ set and \(Z\) a null set.
\(E = V\setminus Z\) with \(V\) a $G_\delta$ set and \(Z\) a null set.
If \(m_*(E)<\infty\), these are also equivalent to:
- For every \(\varepsilon>0\) there exists a compact set \(K\subseteq E\) with \(m_*(E\setminus K) < \varepsilon\).
The finiteness hypothesis in (5) cannot be dropped: \(E=\RR^n\) is measurable, but \(m_*(\RR^n\setminus K)=\infty\) for every compact \(K\subseteq\RR^n\).
Classes of Lebesgue measurable sets (Tag FF-MXNIV)
The following subsets of \(\RR^n\) are Lebesgue measurable:
open sets;
closed sets;
sets \(E\) with outer measure \(m_*(E) = 0\).
If \(M\subseteq\RR^n\) is Lebesgue measurable, then the cylinder \(M\times\RR\subseteq\RR^{n+1}\) is Lebesgue measurable.
Null sets in $\RR^n$ (Tag FD-6HUIM)
Let \(m\) be Lebesgue measure on \(\RR^n\). A set \(A\subseteq\RR^n\) is null if for every \(\varepsilon>0\) there exist open sets \(U_1,U_2,\ldots\subseteq\RR^n\) with \(A\subseteq\bigcup_{j} U_j\) and \(\sum_j m(U_j) < \varepsilon\).
A set \(A\) is null if and only if its outer measure \(m_*(A)\) is \(0\). Indeed, \(m_*(A)\leq\sum_j m(U_j)\) for every such cover; conversely, if \(m_*(A)=0\), a cover of \(A\) by closed cubes of total volume less than \(\varepsilon/2^{n+1}\) can be replaced by the open cubes with the same centers and twice the side lengths, of total volume less than \(\varepsilon\).
Almost disjoint sets (Tag FD-5T3HX)
Let \(X\) be a topological space. Subsets \(A,B\subseteq X\) are almost disjoint if their interiors are disjoint: \(A^\circ \cap B^\circ = \emptyset\).
Almost disjoint sets (Tag FF-NOX4C)
Let \(X\) be a topological space. Subsets \(A, B\subseteq X\) are almost disjoint if and only if \(A^\circ \cap B^\circ = \emptyset\).
Complete measure (Tag FD-T7SN6)
Let \((X,\mcm,\mu)\) be a measure space. The measure \(\mu\) is complete if every subset of a null set is measurable: whenever \(N\in\mcm\), \(\mu(N)=0\), and \(A\subseteq N\), then \(A\in\mcm\).
Convergence in measure (Tag FD-D7HOH)
Let \(m\) be Lebesgue measure on \(\RR^n\), let \(E\subseteq\RR^n\) be Lebesgue measurable, and let \(f_k\colon E\to\RR\) for \(k\geq 1\) and \(f\colon E\to\RR\) be measurable. The sequence \((f_k)\) converges in measure to \(f\) on \(E\) if for every \(\alpha>0\), \[ \lim_{k \to \infty} m\qty{\theset{x \in E \suchthat \abs{f_{k}(x)-f(x)}>\alpha}}=0. \]
The class $L^+$ of nonnegative measurable functions (Tag D-BF5L2)
Let \((X,\mcm)\) be a measurable space. The class \(L^+\), or \(L^+(X,\mcm)\), is the set of measurable functions \(f\colon X\to[0,\infty]\).
Integrals and \(L^p\) spaces
Integrable functions (Tag D-R5DL3)
Let \((X,\mcm,\mu)\) be a measure space. A measurable function \(f\colon X\to\CC\) is integrable if \[ \norm{f}_1 \coloneqq \int_X \abs{f} \dmu < \infty. \]
Essential supremum norm and essentially bounded functions (Tag D-3PVRB)
Let \((X,\mcm,\mu)\) be a measure space and let \(f\colon X\to\CC\) be measurable. The essential supremum of \(\abs{f}\) is \[ \norm{f}_\infty \coloneqq \inf \theset{\alpha \geq 0 \suchthat \mu\qty{\theset{x\in X \suchthat \abs{f(x)} \geq \alpha}} = 0} \in [0,\infty], \] with \(\inf\emptyset=\infty\). The function \(f\) is essentially bounded if there exists a real number \(c\) such that \(\mu\qty{\theset{x\in X \suchthat \abs{f(x)} > c}} = 0\), equivalently \(\norm{f}_\infty < \infty\).
When \(\norm{f}_\infty<\infty\), the set \(\theset{\abs{f}>\norm{f}_\infty}=\bigcup_{k\geq 1}\theset{\abs{f}\geq\norm{f}_\infty+1/k}\) is a countable union of null sets, so \(\abs{f(x)} \leq \norm{f}_\infty\) for almost every \(x\), and \(\norm{f}_\infty\) is the smallest constant with this property.
The $L^\infty$ norm (Tag FD-J5AFR)
Let \((X,\mcm,m)\) be a measure space and let \(f\colon X\to\CC\) be measurable. The \(L^\infty\) norm of \(f\) is \[ \norm{f}_\infty \coloneqq \inf \theset{\alpha \geq 0 \suchthat m\qty{\theset{x\in X\suchthat \abs{f(x)} \geq \alpha}} = 0} \in[0,\infty]. \]
When \(\norm{f}_\infty<\infty\), it is the least real number \(c\) with \(\abs{f}\leq c\) almost everywhere.
$L^\infty$ (Tag D-MFDEJ)
Let \((X,\mcm,\mu)\) be a measure space. The space \(L^\infty(X)\) is \[ L^\infty(X) \coloneqq \theset{f\colon X\to \CC \suchthat f \text{ is measurable and essentially bounded}} = \theset{f\colon X\to \CC \suchthat f \text{ is measurable and } \norm{f}_\infty < \infty}, \] where \(\norm{f}_\infty\) is the essential supremum of \(\abs{f}\).
Convolution of functions on $\RR^n$ (Tag D-TS42Y)
Let \(f,g\colon\RR^n\to\CC\) be measurable. The convolution of \(f\) and \(g\) is the function \[ (f * g)(x) \coloneqq \int_{\RR^n} f(x-y)\, g(y) \, dy, \] defined at every \(x\in\RR^n\) for which the integral converges absolutely.
Convolution $(f\ast g)(\xi) = \int f(\xi - y)g(y) \, dy$ (Tag FF-4WIMZ)
Let \(f, g\colon\RR^n\to\CC\) be measurable. At every \(\xi\in\RR^n\) for which the integral converges absolutely, the convolution of \(f\) and \(g\) is \[ (f\ast g)(\xi) = \int_{\RR^n} f(\xi - y)g(y) \, dy. \]
Fourier transform (Tag D-5LZQ4)
For \(f\in L^1(\RR^n)\), the Fourier transform of \(f\) is the function \(\widehat f\colon\RR^n\to\CC\) given by \[ \widehat f(\xi) \coloneqq \int_{\RR^n} f(x)e^{-2\pi i x\cdot\xi}\,dx, \qquad \xi\in\RR^n. \]
Dilation (Tag D-EWXAT)
Let \(\phi\colon\RR^n\to\CC\) be a function and let \(t>0\). The dilation of \(\phi\) by \(t\) is the function \(\phi_t\colon\RR^n\to\CC\) given by \[ \phi_{t}(x) \coloneqq t^{-n} \phi\qty{t^{-1} x}. \]
Approximations to the identity (Tag D-MAGRK)
Let \(\phi\in L^1(\RR^n)\) with \(\int_{\RR^n} \phi = 1\), and for \(t>0\) let \(\phi_t\) be its dilation \(\phi_t(x)=t^{-n}\phi(t^{-1}x)\). Then \(\phi\), or the family \((\phi_t)_{t>0}\), is called an approximate identity.
For every \(\phi\in L^1(\RR^n)\) and \(t>0\), the substitution \(x=ty\) gives \(\int \phi_{t} = \int \phi\), so every dilation of an approximate identity also has integral \(1\).
Approximations to the identity (Tag D-ARQFC)
Let \(\phi\in L^1(\RR^n)\) with \(\int_{\RR^n}\phi = 1\), and for \(t>0\) let \(\phi_t(x)\coloneqq t^{-n}\phi(t^{-1}x)\) be its dilations. Then \(\phi\), or the family \((\phi_t)_{t>0}\), is called an approximate identity.
Let \(\phi\in L^1(\RR^n)\) with \(\int\phi=1\). Then:
\(\int \phi_t = \int\phi = 1\) and \(\norm{\phi_t}_1 = \norm{\phi}_1\) for every \(t>0\); in particular \(\norm{\phi_t}_1=1\) when \(\phi\geq 0\).
\(\sup_{t>0} \norm{\phi_t}_1 < \infty\).
For every \(h>0\), \(\lim_{t\to 0^+} \int_{\abs{x} \geq h} \abs{\phi_t(x)}\dx = 0\).
The substitution \(x=ty\), \(dx=t^n\,dy\), gives \(\int\phi_t=\int\phi\) and \(\norm{\phi_t}_1=\norm{\phi}_1\), which proves the first two items. The same substitution gives \[ \int_{\abs{x}\geq h}\abs{\phi_t(x)}\dx=\int_{\abs{y}\geq h/t}\abs{\phi(y)}\,dy, \] and the right side tends to \(0\) as \(t\to 0^+\) by dominated convergence, with dominating function \(\abs{\phi}\in L^1\).
Functional analysis
Dual norm (Tag D-T4LOC)
Let \(X\neq\theset{0}\) be a normed vector space over \(\CC\), and let \(\dualof{X}\) be the space of continuous linear functionals \(X\to\CC\). For \(L \in \dualof{X}\), the dual norm or operator norm of \(L\) is \[ \norm{L}_{\dualof{X}} \coloneqq \sup_{ \substack{x\in X \\ \norm{x} = 1} } \abs{L(x)} = \sup_{ \substack{x\in X \\ \norm{x} \leq 1} } \abs{L(x)}. \]
Orthonormal sequence (Tag D-4IXAO)
Let \(H\) be an inner product space with inner product \(\inner{\cdot}{\cdot}\) and norm \(\norm{u}\coloneqq\sqrt{\inner{u}{u}}\), and let \(I\) be a countable index set. A family \((u_i)_{i\in I}\) of elements of \(H\) is orthonormal if
\(\inner{u_i}{u_j} = 0\) for all \(i,j\in I\) with \(i \neq j\), and
\(\norm{u_j} = 1\) for all \(j\in I\).
Basis of a Hilbert space (Tag D-AQX7W)
Let \(\mch\) be a Hilbert space. An orthonormal sequence \((u_n)_{n\geq 1}\) in \(\mch\) is a basis of \(\mch\) if the linear span of \(\theset{u_n \suchthat n\geq 1}\), the set of finite linear combinations of the \(u_n\), is dense in \(\mch\).
Complete families in a Hilbert space (Tag D-SLYE5)
Let \(H\) be a Hilbert space. A family \((u_{n})_{n\geq 1}\) of vectors in \(H\) is complete if the only \(x\in H\) with \(\inner{x}{u_{n}} = 0\) for all \(n\) is \(x = 0\).
Dual of a Hilbert space (Tag D-PQIQO)
Let \(H\) be a Hilbert space over \(\CC\). The dual of \(H\) is the set \(\dualof{H}\) of linear functionals \(L\colon H\to \CC\) that are continuous with respect to the norm of \(H\).
Linear functionals (Tag D-EPSKF)
Let \(X\) be a vector space over \(\CC\). A map \(L\colon X \to \CC\) is a linear functional if \[ L(\alpha x + y) = \alpha L(x) + L(y) \quad \text{for all } x,y\in X \text{ and } \alpha\in\CC. \]
Banach space (Tag D-BG455)
A Banach space is a normed vector space \((V,\norm{\cdot})\) over \(\RR\) or \(\CC\) that is complete as a metric space with the metric \(d(x,y)\coloneqq\norm{x-y}\).
Hilbert space (Tag D-7QQUO)
A Hilbert space is an inner product space \((H,\inner{\cdot}{\cdot})\) over \(\RR\) or \(\CC\) that is a Banach space under the norm \(\norm{x}\coloneqq\sqrt{\inner{x}{x}}\).