Sequential Pointwise Limits and Simple Approximation

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.