Problem 22
Let \(\mu\) be counting measure on \(\mathbb{N}\). Interpret Fatou's lemma and the monotone and dominated convergence theorems as statements about infinite series.
Problem 24
Let \((X, \mathcal{M}, \mu)\) be a measure space with \(\mu(X)<\infty\), and let
\((X, \overline{\mathcal{M}}, \bar{\mu})\) be its completion. Suppose \(f: X
\rightarrow \mathbb{R}\) is bounded. Then \(f\) is
\(\overline{\mathbb{M}}\)-measurable (and hence in
\(\left.L^{1}(\bar{\mu})\right)\) iff there exist sequences
\(\left\\{\phi_{n}\right\\}\) and \(\left\\{\psi_{n}\right\\}\) of
\(\mathcal{M}\)-measurable simple functions such that \(\phi_{n} \leq f \leq
\psi_{n}\) and \(\int\left(\psi_{n}-\phi_{n}\right) d \mu
Problem 25
Let \(f(x)=x^{-1 / 2}\) if \(0
Problem 26
If \(f \in L^{1}(m)\) and \(F(x)=\int_{-\infty}^{x} f(t) d t\), then \(F\) is continuous on \(\mathbb{R}\).
Problem 27
Let \(f_{n}(x)=a e^{-n a x}-b e^{-n b x}\) where \(0
Problem 30
Show that \(\lim _{k \rightarrow \infty} \int_{0}^{k} x^{n}\left(1-k^{-1} x\right)^{k} d x=n !\).
Problem 32
Suppose \(\mu(X)<\infty\). If \(f\) and \(g\) are complex-valued measurable functions on \(X\), define $$ \rho(f, g)=\int \frac{|f-g|}{1+|f-g|} d \mu $$ Then \(\rho\) is a metric on the space of measurable functions if we identify functions that are equal a.e., and \(f_{n} \rightarrow f\) with respect to this metric iff \(f_{n} \rightarrow f\) in measure.
Problem 36
If \(\mu\left(E_{n}\right)<\infty\) for \(n \in \mathbb{N}\) and \(\chi_{E_{n}} \rightarrow f\) in \(L^{1}\), then \(f\) is (a.e. equal to) the characteristic function of a measurable set.
Problem 37
Suppose that \(f_{n}\) and \(f\) are measurable complex-valued functions and \(\phi: \mathbb{C} \rightarrow C\). a. If \(\phi\) is continuous and \(f_{n} \rightarrow f\) a.e., then \(\phi \circ f_{n} \rightarrow \phi \circ f\) a.e. b. If \(\phi\) is uniformly continuous and \(f_{n} \rightarrow f\) uniformly, almost uniformly, or in measure, then \(\phi \circ f_{n} \rightarrow \phi \circ f\) uniformly, almost uniformly, or in measure, respectively. c. There are counterexamples when the continuity assumptions on \(\phi\) are not satisfied.
Problem 38
Suppose \(f_{n} \rightarrow f\) in measure and \(g_{n} \rightarrow g\) in measure. a. \(f_{n}+g_{n} \rightarrow f+g\) in measure. b. \(f_{n} g_{n} \rightarrow f g\) in measure if \(\mu(X)<\infty\), but not necessarily if \(\mu(X)=\infty\).
