Convergence of Measurable Functions

Suppose $X$ is a finite set. Explain why a sequence of functions from $X$ to $\mathbb {R}$ that converges pointwise on $X$ also converges uniformly on $X$.

Give an example of a sequence of functions from $\mathbb {N}$ to $\mathbb {R}$ that converges pointwise on $\mathbb {N}$ but does not converge uniformly on $\mathbb {N}$.

Give an example of a sequence of continuous functions $f_{1}, f_{2}, \ldots$ from $[0,1]$ to $\mathbb {R}$ that converges pointwise to a function $f:[0,1] \rightarrow \mathbb {R}$ that is not a bounded function.

Prove or give a counterexample: If $A \subseteq \mathbb {R}$ and $f_{1}, f_{2}, \ldots$ is a sequence of uniformly continuous functions from $A$ to $\mathbb {R}$ that converges uniformly to a function $f: A \rightarrow \mathbb {R}$, then $f$ is uniformly continuous on $A$.

Give an example to show that Egorov's Theorem can fail without the hypothesis that $\mu (X) < \infty$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space with $\mu (X) < \infty$. Suppose $f_{1}, f_{2}, \ldots$ is a sequence of $\mathcal{S}$-measurable functions from $X$ to $\mathbb {R}$ such that $\lim_{k \rightarrow \infty } f_{k}(x) = \infty$ for each $x \in X$. Prove that for every $\varepsilon > 0$, there exists a set $E \in \mathcal{S}$ such that $\mu (X \backslash E) < \varepsilon$ and $f_{1}, f_{2}, \ldots$ converges uniformly to $\infty$ on $E$ (meaning that for every $t > 0$, there exists $n \in \mathbb {N}$ such that $f_{k}(x) > t$ for all integers $k \geq n$ and all $x \in E$).

Suppose $F$ is a closed bounded subset of $\mathbb {R}$ and $g_{1}, g_{2}, \ldots$ is an increasing sequence of continuous real-valued functions on $F$ (thus $g_{1}(x) \leq g_{2}(x) \leq \cdots$ for all $x \in F$) such that $\sup \left\{ g_{1}(x), g_{2}(x), \ldots \right\} < \infty$ for each $x \in F$. Define a real-valued function $g$ on $F$ by

Prove that $g$ is continuous on $F$ if and only if $g_{1}, g_{2}, \ldots$ converges uniformly on $F$ to $g$.

Suppose $\mu$ is the measure on $\left(\mathbb {N}, 2^{\mathbb {N}}\right)$ defined by

Prove that for every $\varepsilon > 0$, there exists a set $E \subseteq \mathbb {N}$ with $\mu \left(\mathbb {N} \backslash E\right) < \varepsilon$ such that $f_{1}, f_{2}, \ldots$ converges uniformly on $E$ for every sequence of functions $f_{1}, f_{2}, \ldots$ from $\mathbb {N}$ to $\mathbb {R}$ that converges pointwise on $\mathbb {N}$.

Suppose $F_{1}, \ldots , F_{n}$ are disjoint closed subsets of $\mathbb {R}$. Prove that if

is a function such that $\left.g\right|_{F_{k}}$ is a continuous function for each $k \in \left\{ 1, \ldots , n\right\}$, then $g$ is a continuous function.

Suppose $F \subseteq \mathbb {R}$ is such that every continuous function from $F$ to $\mathbb {R}$ can be extended to a continuous function from $\mathbb {R}$ to $\mathbb {R}$. Prove that $F$ is a closed subset of $\mathbb {R}$.

Prove or give a counterexample: If $F \subseteq \mathbb {R}$ is such that every bounded continuous function from $F$ to $\mathbb {R}$ can be extended to a continuous function from $\mathbb {R}$ to $\mathbb {R}$, then $F$ is a closed subset of $\mathbb {R}$.

Give an example of a Borel measurable function $f$ from $\mathbb {R}$ to $\mathbb {R}$ such that there does not exist a set $B \subseteq \mathbb {R}$ such that $\left|\mathbb {R} \backslash B\right| = 0$ and $\left.f\right|_{B}$ is a continuous function on $B$.

Prove or give a counterexample: If $f_{t}: \mathbb {R} \rightarrow \mathbb {R}$ is a Borel measurable function for each $t \in \mathbb {R}$ and $f: \mathbb {R} \rightarrow (-\infty , \infty ]$ is defined by

then $f$ is a Borel measurable function.

Suppose $b_{1}, b_{2}, \ldots$ is a sequence of real numbers. Define $f: \mathbb {R} \rightarrow [0, \infty ]$ by

Prove that $\left|\left\{ x \in \mathbb {R}: f(x) < 1\right\} \right| = \infty$.

Suppose $B$ is a Borel set and $f: B \rightarrow \mathbb {R}$ is a Lebesgue measurable function. Show that there exists a Borel measurable function $g: B \rightarrow \mathbb {R}$ such that