The Riesz–Fischer Theorem

In the settings of Measurable Functions and the Lebesgue Integral: Standing Notation and Real Normed, Inner Product and Hilbert Spaces: Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $p\ge1$ be a real number. $\mathcal{L}^{p}(\mu)$ is the set of $p$-integrable functions, with the $p$-seminorm $\|f\|_{p}$, and $L^{p}(\mu)$ is the Lebesgue space, with the classes $[f]$ of The Lebesgue Space Lᵖ(μ) (i), the addition and scalar multiplication of The Lebesgue Space Lᵖ(μ) (ii) and the norm $\|\cdot\|_{p}$ of The Lebesgue Space Lᵖ(μ) (iii).

(i) A real normed space. $L^{p}(\mu)$ is a real vector space, whose zero vector is the class $[0]$ of the zero function and in which

$$[f]-[g]=[f-g]\qquad(f,g\in\mathcal{L}^{p}(\mu));$$

and $\|\cdot\|_{p}$ is a norm on it, so that $(L^{p}(\mu),\|\cdot\|_{p})$ is a real normed space.

For the next two clauses, let $(f_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{L}^{p}(\mu)$ that is Cauchy for the $p$-seminorm: for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\|f_{m}-f_{k}\|_{p}<\varepsilon$ for all $m,k\in\mathbb{N}$ with $m\ge N$ and $k\ge N$. Then there are $f\in\mathcal{L}^{p}(\mu)$, a strictly increasing sequence $(m_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ and $g\in\mathcal{L}^{p}(\mu)$ such that the following hold.

(ii) Convergence in the $p$-seminorm.

$$\|f_{m}-f\|_{p}\to0 .$$

(iii) A dominated subsequence converging almost everywhere. For $\mu$-almost every $x\in X$,

$$f_{m_{k}}(x)\to f(x)\qquad\text{and}\qquad|f_{m_{k}}(x)|\le g(x)\ \text{ for every }k\in\mathbb{N}.$$

(iv) Completeness. Assume the axiom of dependent choice. Then $(L^{p}(\mu),\|\cdot\|_{p})$ is a real Banach space.