Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space. Measurable maps from $X$ to $\mathbb{R}$ are as in Measurable Map, and the Set of Measurable Maps (iii). Sequences are as in Sequences (i), and a sequence of measurable maps $X\to\mathbb{R}$ is a sequence in the set $\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ of such maps of Measurable Map, and the Set of Measurable Maps (ii). Let $(h_{m})_{m\in\mathbb{N}}$ be a sequence of measurable maps $X\to\mathbb{R}$.

(i) If for every $x\in X$ the set $\{h_{m}(x):m\in\mathbb{N}\}$ is bounded above in $\mathbb{R}$, then the map $x\mapsto\sup_{m}h_{m}(x)$, with the supremum of that set formed in $\mathbb{R}$, from $X$ to $\mathbb{R}$ is measurable. If for every $x\in X$ that set is bounded below in $\mathbb{R}$, then the map $x\mapsto\inf_{m}h_{m}(x)$, with the infimum formed in $\mathbb{R}$, is measurable.

(ii) If for every $x\in X$ the sequence $(h_{m}(x))_{m\in\mathbb{N}}$ is convergent, then the map $x\mapsto\lim_{m\to\infty}h_{m}(x)$, with the limit of The Limit of a Convergent Sequence (i), from $X$ to $\mathbb{R}$ is measurable.

(iii) The set of those $x\in X$ for which $(h_{m}(x))_{m\in\mathbb{N}}$ is convergent belongs to $\mathcal{F}$.