Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions

Let $(X,\mathcal{F})$ be a measurable space. Real-valued maps on $X$ are called measurable when they are measurable with respect to $\mathcal{F}$ and the Borel $\sigma$-algebra on the real line.

Let $K$ be a real number and let $f_n:X\to\mathbb{R}$, indexed by the natural numbers $n$, be measurable maps with $|f_n(x)|\le K$ for all $n$ and all $x\in X$.

1. (Countable suprema and infima.) The maps $g(x)=\sup_n f_n(x)$ and $h(x)=\inf_n f_n(x)$ take real values and are measurable.

2. (Bounded pointwise limits.) Suppose in addition that for every $x\in X$ the sequence $(f_n(x))_{n\in\mathbb{N}}$ converges to a real number, denoted $f(x)$. Then $f:X\to\mathbb{R}$ is measurable.

3. (Monotone functions.) Let $a<b$ be real numbers and let $u:[a,b]\to\mathbb{R}$ be nondecreasing, meaning that $u(s)\le u(t)$ whenever $a\le s\le t\le b$. Then $u$ is measurable with respect to the trace Borel $\sigma$-algebra on $[a,b]$ and the Borel $\sigma$-algebra on the real line.

4. (Continuous functions.) Let $a<b$ be real numbers, and regard $[a,b]$ as a subset of the real line with the absolute value metric. Every map $u:[a,b]\to\mathbb{R}$ that is continuous on $[a,b]$ is measurable with respect to the trace Borel $\sigma$-algebra on $[a,b]$ and the Borel $\sigma$-algebra on the real line.