The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $p$ be a real number with $1\le p$. Write $\mathcal{L}^{p}$ for the set of $p$-integrable functions on $(X,\mathcal{F},\mu)$, and let $L^{p}=L^{p}(X,\mathcal{F},\mu)$ be the Lebesgue space of classes of such functions, carrying the operations defined there and the number $\lVert[f]\rVert_{p}$ of The Lebesgue Space of Power-Integrable Functions §norm. Then the following hold.

1. (A real normed space)¶ $L^{p}$, with those operations, is a vector space over $\mathbb{R}$ whose zero vector is the class of the map taking the value $0$ at every point of $X$, and $[f]\mapsto\lVert[f]\rVert_{p}$ is a norm on it. Thus $L^{p}$ is a real normed space, with distance given by $d_{p}([f],[g])=\lVert[f-g]\rVert_{p}$.

2. (Completeness)¶ $L^{p}$ is a real Banach space.

3. (An almost-everywhere convergent subsequence with a dominating function)¶ Let $(F_{m})_{m\in\mathbb{N}}$ be a sequence in $L^{p}$ converging to $F\in L^{p}$ in the metric of claim 1. For each $m\in\mathbb{N}$ let $f_{m}\in\mathcal{L}^{p}$ be a representative of $F_{m}$, and let $f\in\mathcal{L}^{p}$ be a representative of $F$. Then there are natural numbers $m_{1}<m_{2}<m_{3}<\dots$, a function $h\in\mathcal{L}^{p}$ and a null set $N$ such that for every $x\in X\setminus N$ the sequence $(f_{m_{j}}(x))_{j\in\mathbb{N}}$ converges to $f(x)$ and