Chapter 2

Suppose \(f, g: X \rightarrow \mathbb{R}\) are measurable. a. \(f g\) is measurable (where \(0 \cdot(\pm \infty)=0\) ). b. Fix \(a \in \overline{\mathbb{R}}\) and define \(h(x)=a\) if \(f(x)=-g(x)=\pm \infty\) and \(h(x)=\) \(f(x)+g(x)\) otherwise. Then \(h\) is measurable.

If \(f: X \rightarrow \mathbb{R}\) and \(f^{-1}((r, \infty]) \in \mathcal{M}\) for each \(r \in \mathbb{Q}\), then \(f\) is measurable.

If \(X=A \cup B\) where \(A, B \in \mathcal{M}\), a function \(f\) on \(X\) is measurable iff \(f\) is measurable on \(A\) and on \(B\).

If \(f: \mathbb{R} \rightarrow \mathbb{R}\) is monotone, then \(f\) is Borel measurable.

Suppose that \(f\) is a function on \(\mathbb{R} \times \mathbb{R}^{k}\) such that \(f(x, \cdot)\) is Borel measurable for each \(x \in \mathbb{R}\) and \(f(\cdot, y)\) is continuous for each \(y \in \mathbb{R}^{k}\). For \(n \in \mathbb{N}\), define \(f_{n}\) as follows. For \(i \in \mathbb{Z}\) let \(a_{i}=i / n\), and for \(a_{i} \leq x \leq a_{i+1}\) let $$ f_{n}(x, y)=\frac{f\left(a_{i+1}, y\right)\left(x-a_{i}\right)-f\left(a_{i}, y\right)\left(x-a_{i+1}\right)}{a_{i+1}-a_{i}} $$ Then \(f_{n}\) is Borel measurable on \(\mathbb{R} \times \mathbb{R}^{k}\) and \(f_{n} \rightarrow f\) pointwise; hence \(f\) is Borel measurable on \(\mathbb{R} \times \mathbb{R}^{k}\). Conclude by induction that every function on \(\mathbb{R}^{n}\) that is continuous in each variable separately is Borel measurable.

Suppose \(\left\\{f_{n}\right\\} \subset L^{+}, f_{n} \rightarrow f\) pointwise, and \(\int f=\lim \int f_{n}<\infty\). Then \(\int_{E} f=\lim \int_{E} f_{n}\) for all \(E \in \mathcal{M}\). However, this need not be true if \(\int f=\lim \int f_{n}=\) \(\infty\).

If \(f \in L^{+}\), let \(\lambda(E)=\int_{E} f d \mu\) for \(E \in \mathcal{M}\). Then \(\lambda\) is a measure on \(\mathcal{M}\), and for any \(g \in L^{+}, \int g d \lambda=\int f g d \mu\). (First suppose that \(g\) is simple.)

If \(\left\\{f_{n}\right\\} \subset L^{+}, f_{n}\) decreases pointwise to \(f\), and \(\int f_{1}<\infty\), then \(\int f=\lim \int f_{n}\).

Assume Fatou's lemma and deduce the monotone convergence theorem from it.

Fatou's lemma remains valid if the hypothesis that \(f_{n} \in L^{+}\)is replaced by the hypothesis that \(f_{n}\) is measurable and \(f_{n} \geq-g\) where \(g \in L^{+} \cap L^{1}\). What is the analogue of Fatou's lemma for nonpositive functions?