Lebesgue Measurable Functions

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.

Let $\left\{ f_{n}:[a, b] \rightarrow \mathbb {R}\right\}$ be an increasing sequence of continuous functions that converges pointwise to the continuous function $f:[a, b] \rightarrow \mathbb {R}$. Show that the convergence is uniform on $[a, b]$.

A real-valued measurable function is said to be semisimple provided that it takes only a countable number of values. Let $f: E \rightarrow \mathbb {R}$ be measurable. Show that there is a sequence of semisimple functions that converges to $f$ uniformly on $E$.

Assume that $m(E) < \infty$ and let $f: E \rightarrow \mathbb {R}$ be measurable. Show that for each $\epsilon > 0$, there is a measurable subset $E_{0}$ of $E$ such that $f$ is bounded on $E_{0}$ and $m\left(E \sim E_{0}\right) < \epsilon$.

Assume that $m(E) < \infty$ and let $f: E \rightarrow \mathbb {R}$ be measurable. Show that for each $\epsilon > 0$, there is a measurable subset $E_{0}$ of $E$ and a sequence $\left\{ \varphi : E \rightarrow \mathbb {R}\right\}$ of simple functions for which $\left\{ \varphi_{n}\right\} \rightarrow f$ uniformly on $E_{0}$ and $m\left(E \sim E_{0}\right) < \epsilon$.

Let $E$ be a measurable subset of $[a, b]$, and define $f:[a, b] \rightarrow \mathbb {R}$ by $f = \chi_{E}$. For each $\epsilon > 0$, show that there is a measurable subset $E_{0} \subseteq [a, b]$ and step-function $h:[a, b] \rightarrow \mathbb {R}$ for which

Let $A$ and $B$ be any sets. Show that

For a sequence $\left\{ f_{n}: E \rightarrow \mathbb {R}\right\}$ of measurable functions, show that the functions $\inf \left\{ f_{n}\right\}$ and $\sup \left\{ f_{n}\right\}$ are measurable.

For $f:[a, b] \rightarrow \mathbb {R}$ measurable, let $E_{+} = \left\{ x \in E \mid f(x) \geq 0\right\}$ and $E_{-} = \left\{ x \in E \mid f(x) < 0\right\}$. By considering the restriction of $f$ to $E_{+}$and $E_{-}$, prove the general Simple Approximation Theorem based on the special case of a non-negative measurable function.

Let $f:[a, b] \rightarrow \mathbb {R}$ be increasing. Show that $f$ is measurable by first showing that, for each $n$, the strictly increasing function $x \mapsto f(x)+x / n$ is measurable, and then taking pointwise limits.

Verify the continuity of the function $f: \mathbb {R} \rightarrow \mathbb {R}$ defined in the proof of Lemma 10.

Lemma 10: If $F$ is a closed subset of $\mathbb {R}$ and $f: F \rightarrow \mathbb {R}$ is a continuous function, then it has a continuous extension to $f: \mathbb {R} \rightarrow \mathbb {R}$. The proof constructs this extension as follows: $\mathbb {R} \sim F$ is open, so $\mathbb {R} \sim F = \bigcup_{n=1}^{\infty } I_{n}$, a countable, disjoint union of open intervals. If $I_{n} = (a_{n}, b_{n})$ is bounded, define $f$ on $\left[a_{n}, b_{n}\right]$ to be any continuous function that agrees with $f$ at the end-points. If $I_{n}$ is unbounded, define $f$ on $I_{n}$ to be the constant function that agrees with $f$ at the finite end-point of $I_{n}$.

For the function $f: E \rightarrow \mathbb {R}$ and the set $F$ in the statement of Lusin's Theorem, show that the restriction of $f$ to $F$ is continuous. Must there be any points in $E$ at which $f: E \rightarrow \mathbb {R}$ is continuous?

Show that the conclusion of Egoroff's Theorem can fail if we drop the assumption that the domain has finite measure.

Show that Egoroff's Theorem continues to hold if $f$ and each $f_{n}$ is an extended real-valued function that is finite almost everywhere, and the convergence is pointwise almost everywhere on $E$.

Let $\left\{ f_{n}: E \rightarrow \mathbb {R}\right\}$ be a sequence of measurable functions that converges pointwise on $E$ to $f: E \rightarrow \mathbb {R}$. Show that $E = \bigcup_{k = 1}^{\infty } E_{k}$, where for each $E_{k}$ is measurable, and $\left\{ f_{n}\right\}$ converges uniformly to $f$ on each $E_{k}$ if $k > 1$, and $m\left(E_{1}\right) = 0$.

Show that in Corollary 11, if $f$ is bounded, then the sequence $\left\{ f_{n}\right\}$ may be chosen to be uniformly bounded.

Corollary 11: If $f: E \rightarrow \mathbb {R}$ is a measurable function, then there is a sequence of continuous functions $\left\{ f_{n}: \mathbb {R} \rightarrow \mathbb {R}\right\}$ that converges pointwise almost everywhere on $E$ to $f$.