5 problems.
- \(C([0,1])\) is complete in the uniform norm but not in the \(L^1\) norm
- Continuity from above for a finite Borel measure, and \(\varepsilon\)-\(\delta\) absolute continuity
- If \(f_k\to f\) a.e. with \(\|f_k\|_2\leq M\) on \([0,1]\), then \(f\in L^2\), \(\|f\|_2\leq M\), and \(\int f_k\to\int f\)
- The subgraph of a nonnegative function is measurable if and only if the function is, and \(m(\mathcal{A})=\int_{\RR^n}f=\int_0^\infty m(\{f\ge t\})\,dt\)
- \(L^2([0,1])\) is dense in \(L^1([0,1])\), and Riesz representation for \(L^1([0,1])\)