Sums, Products, and Compositions

Let $A$ and $B$ be measurable sets. Is $f: A \cup B \rightarrow \mathbb {R}$ continuous if and only if its restrictions to $A$ and $B$ are continuous?

Show that if $h: E \rightarrow \mathbb {R}$ is measurable, then for any $\alpha$, so is $\alpha \cdot h: E \rightarrow \mathbb {R}$.

Provide an example of a function $f:[a, b] \rightarrow \mathbb {R}$ that is not measurable, while both $\left|f\right|$ and $f^{2}$ are measurable.

Let $\left\{ f_{n}: E \rightarrow \mathbb {R}\right\}$ be a sequence of measurable functions. Define $E_{0}$ to be the set of points $x$ in $E$ at which $\left\{ f_{n}(x)\right\}$ converges to a real number. Is the set $E_{0}$ measurable?

Let $E$ be measurable and $f: E \rightarrow \mathbb {R}$ be continuous except at a countable number of points. Show that $f$ is measurable.

If $E$ is measurable and the function $f: E \rightarrow \mathbb {R}$ has the property that $f^{-1}(c)$ is measurable for each number $c$, is $f$ necessarily measurable?

If the function $f: E \rightarrow \mathbb {R}$ has the property that $\left\{ x \in E \mid f(x) > c\right\}$ is a measurable set for each rational number $c$, is $f$ necessarily measurable?

Show that $f: E \rightarrow \mathbb {R}$ is measurable if and only if the function $\hat{f}: \mathbb {R} \rightarrow \mathbb {R}$ is measurable, where $\hat{f}(x) = f(x)$ for $x \in E$ and $\hat{f}(x) = 0$ for $x \notin E$.

Show that $f: E \rightarrow \mathbb {R}$ is measurable if and only if for each Borel set $A, f^{-1}(A)$ is measurable.

A function $f: E \rightarrow \mathbb {R}$ is said to be Borel measurable provided that its domain $E$ is a Borel set and for each $c$, the set $\left\{ x \in E \mid f(x) < c\right\}$ is a Borel set. Verify that Proposition 1 and Theorem 5 remain valid if we replace "(Lebesgue) measurable set" by "'Borel set."

Proposition 1 (Chapter 3): If $f: E \rightarrow \mathbb {R}$ is measurable, then for every interval $I$ of real numbers, $f^{-1}(I)$ is measurable. Theorem 5 (Chapter 3): If $f: E \rightarrow \mathbb {R}$ and $g: E \rightarrow \mathbb {R}$ are measurable functions, then for any $\alpha$ and $\beta$, $\alpha f + \beta g: E \rightarrow \mathbb {R}$ is measurable (linearity), and $f \cdot g: E \rightarrow \mathbb {R}$ is measurable (products). Verify the following:

Assume that $f: \mathbb {R} \rightarrow \mathbb {R}$ is measurable and $g: \mathbb {R} \rightarrow \mathbb {R}$ is continuous. Is the composition $f \circ g$ necessarily measurable?

Suppose that $f:[a, b] \rightarrow R$ is strictly increasing, continuous function that has a Lipschitz inverse. Show that $g \circ f:[a, b] \rightarrow R$ is measurable whenever $g: R \rightarrow R$ is measurable.