\(\star\): Show that for \(E\subseteq \RR^n\), TFAE:
- \(E\) is measurable
- \(E = H\union Z\) here \(H\) is \(F_\sigma\) and \(Z\) is null
- \(E = V\setminus Z'\) where \(V\in G_\delta\) and \(Z'\) is null.
\(\star\): Show that if \(E\subseteq \RR^n\) is measurable then \(m(E) = \sup \theset{ m(K) \suchthat K\subset E\text{ compact}}\) iff for all \(\eps> 0\) there exists a compact \(K\subseteq E\) such that \(m(K) \geq m(E) - \eps\).
\(\star\): Show that cylinder functions are measurable, i.e. if \(f\) is measurable on \(\RR^s\), then \(F(x, y) \definedas f(x)\) is measurable on \(\RR^s\cross \RR^t\) for any \(t\).
\(\star\): Prove that the Lebesgue integral is translation invariant, i.e. if \(\tau_h(x) = x+h\) then \(\int \tau_h f = \int f\).
\(\star\): Prove that the Lebesgue integral is dilation invariant, i.e. if \(f_\delta(x) = {f({x\over \delta}) \over \delta^n}\) then \(\int f_\delta = \int f\).
\(\star\): Prove continuity in \(L^1\), i.e. \[ f \in L^{1} \Longrightarrow \lim _{h \rightarrow 0} \int|f(x+h)-f(x)|=0 .\]
\(\star\): Show that \[f,g \in L^1 \implies f\ast g \in L^1 \qtext{and} \norm{f\ast g}_1 \leq \norm{f}_1 \norm{g}_1.\]
\(\star\): Show that if \(X\subseteq \RR\) with \(\mu(X) < \infty\) then \[ \norm{f}_p \converges{p\to\infty}\to \norm{f}_\infty .\]
Solution
Let \(m\) be Lebesgue measure.
1. \(E \subseteq \RR^n\) is measurable if and only if \(E = H \cup Z\) with \(H\) an \(F_\sigma\) set and \(Z\) null, if and only if \(E = V \setminus Z'\) with \(V\) a \(G_\delta\) set and \(Z'\) null.
If \(E\) is measurable, outer regularity gives open \(G_k \supseteq E\) and, applied to \(E^c\), closed \(F_k \subseteq E\) with \(m(G_k \setminus E) < 1/k\) and \(m(E \setminus F_k) < 1/k\). Then \(H = \bigcup_k F_k\) and \(V = \bigcap_k G_k\) satisfy \(m(E \setminus H) = 0\) and \(m(V \setminus E) = 0\). Conversely \(F_\sigma\) and \(G_\delta\) sets are Borel and null sets are measurable. See E-D75O4.
2. \(m(E) = \sup\theset{m(K) : K \subseteq E \text{ compact}}\) if and only if for every \(\eps > 0\) there is a compact \(K \subseteq E\) with \(m(K) \ge m(E) - \eps\).
\(m(E)\) is an upper bound of \(\theset{m(K) : K \subseteq E \text{ compact}}\) by monotonicity. It is the least upper bound exactly when no \(m(E) - \eps\) with \(\eps > 0\) is an upper bound, which is the stated condition. See E-3OAGD.
3. If \(f\) is Lebesgue measurable on \(\RR^s\), then \(F(x,y) = f(x)\) is Lebesgue measurable on \(\RR^s \times \RR^t\).
1. Then \(V \times \RR^t\) is Borel and \(Z \times \RR^t\) is null, so \(F^{-1}(W)\) is measurable. See E-JJ746.
4. \(\int f(x+h)\,dx = \int f\) and \(\int \delta^{-n}f(x/\delta)\,dx = \int f\) for \(f \geq 0\) measurable or \(f \in L^1\).
For \(f = \chi_E\) these are \(m(E - h) = m(E)\) and \(\delta^{-n}m(\delta E) = m(E)\). Linearity extends them to simple functions, the monotone convergence theorem to measurable \(f \geq 0\), and the decomposition \(f = f^+ - f^-\) to \(L^1\). See E-QFOPP and E-KVDIA.
5. For \(f \in L^1(\RR^n)\), \(\lim_{h\to 0}\int|f(x+h) - f(x)|\,dx = 0\).
For the indicator of a bounded box \(I\), the integral is \(m(I \triangle (I - h))\), which tends to \(0\). Finite linear combinations of such indicators are dense in \(L^1\), and by step 4 translation preserves \(L^1\) distances, so for such a combination \(s\) with \(\|f - s\|_1 < \eps\), \(\int|f(x+h) - f(x)|\,dx < 2\eps + \int|s(x+h) - s(x)|\,dx\). See E-F6IXU.
6. If \(f, g \in L^1\), then \(f \ast g \in L^1\) and \(\|f \ast g\|_1 \le \|f\|_1\|g\|_1\).
By Tonelli’s theorem and step 4, \(\iint |f(x-y)||g(y)|\,dy\,dx = \|f\|_1\|g\|_1 < \infty\). So \(f\ast g(x)\) is defined for a.e. \(x\) by Fubini’s theorem, and \(\int|f\ast g| \le \iint |f(x-y)||g(y)|\,dy\,dx\). See E-DUWXR.
7. If \(\mu(X) < \infty\), then \(\|f\|_p \to \|f\|_\infty\) as \(p \to \infty\).
If \(\mu(X) = 0\) both sides vanish. Otherwise \(\|f\|_p \le \|f\|_\infty\,\mu(X)^{1/p} \to \|f\|_\infty\), and for \(0 < M < \|f\|_\infty\) the set \(\theset{|f| > M}\) has positive finite measure, so \(\|f\|_p \ge M\mu\theset{|f|>M}^{1/p} \to M\). See E-TVBUC.
8. Q.E.D.
Steps <1>1–<1>7 treat the parts in order.