Proof of The Riesz–Fischer Theorem

Throughout, every element of $L^{p}(\mu)$ is $[f]$ for some $f\in\mathcal{L}^{p}(\mu)$, and $[f]=[g]$ if and only if $f(x)=g(x)$ for almost every $x$ (Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (iv), Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (v), Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (i), Almost-Everywhere Equality on the p-Integrable Functions (i)).

A real normed space. The zero function $0$ lies in $\mathcal{L}^{p}(\mu)$: $|0|^{p}$ is the zero function by Real Powers (ii), with integral $0$ by Basic Properties of the Integral of Nonnegative Measurable Maps (v). Let $f,g,h\in\mathcal{L}^{p}(\mu)$ and $\lambda,\nu\in\mathbb{R}$. By The Lebesgue Space Lᵖ(μ) (ii), each vector-space law for classes is the class of the corresponding law for functions, which holds pointwise; for instance

$$([f]+[g])+[h]=[(f+g)+h]=[f+(g+h)]=[f]+([g]+[h]),\qquad \lambda([f]+[g])=[\lambda f+\lambda g]=\lambda[f]+\lambda[g],$$

and in the same way $[f]+[g]=[g]+[f]$, $\lambda(\nu[f])=(\lambda\nu)[f]$, $1[f]=[f]$ and $(\lambda+\nu)[f]=\lambda[f]+\nu[f]$. Since $[f]+[0]=[f+0]=[f]$, the class $[0]$ satisfies Vector Space over a Field (iii). If $z\in L^{p}(\mu)$ satisfies $F+z=F$ for every $F\in L^{p}(\mu)$, then $z=z+[0]=[0]+z=[0]$, and $[f]+[-f]=[f+(-f)]=[0]=z$; this is Vector Space over a Field (iv). Hence $L^{p}(\mu)$ is a real vector space (Vector Space over a Field (ix), Vector Space over a Field (x)), with zero vector $[0]$ and with $-[f]=[-f]$ (The Zero Vector, Negatives and Differences in a Vector Space (i), The Zero Vector, Negatives and Differences in a Vector Space (ii)). By The Zero Vector, Negatives and Differences in a Vector Space (iii),

$$[f]-[g]=[f]+[-g]=[f+(-g)]=[f-g].$$

For the norm, $\|[f]\|_{p}=\|f\|_{p}\ge0$ by Power-Integrable Functions and the p-Seminorm (iii) and Properties of Real Powers (ii). If $\|[f]\|_{p}=0$, then $f(x)=0$ for almost every $x$ by Elementary Properties of the p-Seminorm (vii), so $[f]=[0]$. By Elementary Properties of the p-Seminorm (iii) and Minkowski's Inequality (ii),

$$\|\lambda[f]\|_{p}=\|\lambda f\|_{p}=|\lambda|\,\|[f]\|_{p},\qquad \|[f]+[g]\|_{p}=\|f+g\|_{p}\le\|[f]\|_{p}+\|[g]\|_{p}.$$

These are the conditions of Norms and Real Normed Spaces (v), so $(L^{p}(\mu),\|\cdot\|_{p})$ is a real normed space (Norms and Real Normed Spaces (vi)). This proves (i).

For (ii) and (iii), let $(f_{m})$ be Cauchy for the $p$-seminorm. For $j\in\mathbb{N}_{0}$ write $2^{-j}$ for the power $(1/2)^{j}$; thus $2^{-j}>0$, $2^{0}=1$ and, for $k\in\mathbb{N}$,

$$2^{1-k}=2\cdot2^{-k},\qquad 2^{-(k+1)}=\tfrac12\,2^{-k}<2^{-k}.$$

Step 1: a fast subsequence. For $k\in\mathbb{N}$ let $A_{k}$ be the set of $n\in\mathbb{N}$ such that $\|f_{m}-f_{l}\|_{p}<2^{-k}$ for all $m,l\ge n$. By hypothesis $A_{k}$ has an element, so it has a least element $n_{k}$ (Arithmetic and Order of the Natural Numbers (xv)); no choice is involved. Since $2^{-(k+1)}<2^{-k}$, we have $A_{k+1}\subseteq A_{k}$ and so $n_{k}\le n_{k+1}$. Put $m_{k}=n_{k}+k$. Then $m_{k}<m_{k+1}$ and $m_{k}\ge n_{k}$ (Arithmetic and Order of the Natural Numbers (ix), Arithmetic and Order of the Natural Numbers (xi)), so $(m_{k})$ is strictly increasing (Monotone Sequences (i)), and since $m_{k+1}>m_{k}\ge n_{k}$,

$$\|f_{m_{k+1}}-f_{m_{k}}\|_{p}<2^{-k}\qquad(k\in\mathbb{N}).$$

Step 2: the dominating function. Put $d_{k}=f_{m_{k+1}}-f_{m_{k}}$, an element of $\mathcal{L}^{p}(\mu)$ by Minkowski's Inequality (i), and let $G_{1}=|f_{m_{1}}|$ and $G_{k+1}=|f_{m_{1}}|+\sum_{j=1}^{k}|d_{j}|$, so that $G_{k+1}=G_{k}+|d_{k}|$ and $0\le G_{k}\le G_{k+1}$. Let $C=\|f_{m_{1}}\|_{p}+1$. We show by induction on $k$ (Arithmetic and Order of the Natural Numbers (xiii)) that

$$G_{k}\in\mathcal{L}^{p}(\mu),\qquad \|G_{k}\|_{p}\le C-2^{1-k},\qquad |f_{m_{k}}(x)|\le G_{k}(x)\quad(x\in X).$$

For $k=1$ this holds by Elementary Properties of the p-Seminorm (iv), as $2^{0}=1$. If it holds for $k$, then $|d_{k}|\in\mathcal{L}^{p}(\mu)$ with $\||d_{k}|\|_{p}=\|d_{k}\|_{p}$ (Elementary Properties of the p-Seminorm (iv)), so $G_{k+1}\in\mathcal{L}^{p}(\mu)$ (Minkowski's Inequality (i)) and, by Minkowski's Inequality (ii) and Step 1,

$$\|G_{k+1}\|_{p}\le\|G_{k}\|_{p}+\|d_{k}\|_{p}<C-2\cdot2^{-k}+2^{-k}=C-2^{-k};$$

and $|f_{m_{k+1}}(x)|\le|f_{m_{k}}(x)|+|d_{k}(x)|\le G_{k+1}(x)$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (xi). In particular $\|G_{k}\|_{p}\le C$ for every $k$.

The functions $|G_{k}|^{p}$ lie in $\mathcal{M}^{+}$ (Power-Integrable Functions and the p-Seminorm (i)), form a pointwise nondecreasing sequence (Properties of Real Powers (vi)), and satisfy $\int|G_{k}|^{p}\,d\mu=\|G_{k}\|_{p}^{p}\le C^{p}$ (Elementary Properties of the p-Seminorm (ii), Properties of Real Powers (vi)). Let $H:X\to[0,\infty]$ be $H(x)=\sup_{k}|G_{k}(x)|^{p}$ (Order and Suprema in the Extended Real Line (iii)). By Monotone Convergence Theorem (i) and Monotone Convergence Theorem (ii), $H\in\mathcal{M}^{+}$ and

$$\int H\,d\mu=\sup_{k}\int|G_{k}|^{p}\,d\mu\le C^{p}<\infty .$$

By The Lebesgue Integral and Null Sets (v) the set $Z=\{x\in X:H(x)=\infty\}$ belongs to $\mathcal{F}$ and $\mu(Z)=0$; so does $X\setminus Z$ belong to $\mathcal{F}$ (σ-Algebra and Measurable Space (i)). For $x\in X\setminus Z$, $H(x)$ is a real number and $|G_{k}(x)|^{p}\le H(x)$, hence, by Properties of Real Powers (vii) and Properties of Real Powers (vi),

$$G_{k}(x)\le H(x)^{1/p}\qquad(k\in\mathbb{N},\ x\in X\setminus Z).$$

Let $g:X\to\mathbb{R}$ be $g(x)=\sup_{k}G_{k}(x)$ for $x\in X\setminus Z$ and $g(x)=0$ for $x\in Z$. For $A\in\mathcal{F}$ the indicator function $\mathbf{1}_{A}$ is measurable, its superlevel sets being $\emptyset$, $A$ or $X$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v)). So $g$ is the pointwise supremum of the measurable functions $G_{k}\mathbf{1}_{X\setminus Z}$ (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i)), which are bounded above at every point, so $g$ is measurable by Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps (i). Since $0\le g(x)\le H(x)^{1/p}$ off $Z$ and $g=0$ on $Z$, we get $|g|^{p}\le H$ (Properties of Real Powers (vi), Properties of Real Powers (vii), Real Powers (ii)), so $\int|g|^{p}\,d\mu\le\int H\,d\mu<\infty$ by Basic Properties of the Integral of Nonnegative Measurable Maps (iii), and $g\in\mathcal{L}^{p}(\mu)$. Moreover

$$|f_{m_{k}}(x)|\le G_{k}(x)\le g(x)\qquad(k\in\mathbb{N},\ x\in X\setminus Z).$$

Step 3: the limit function. Let $x\in X\setminus Z$. The partial sums $\sum_{j=1}^{k}|d_{j}(x)|=G_{k+1}(x)-|f_{m_{1}}(x)|$ are bounded above by $g(x)$, so the series $\sum_{j}|d_{j}(x)|$ converges (Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (i)); hence $\sum_{j}d_{j}(x)$ converges absolutely and so converges (Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (iii)), and as $d_{j}(x)=f_{m_{j+1}}(x)-f_{m_{j}}(x)$, the sequence $(f_{m_{k}}(x))_{k}$ is convergent (Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping (vii)).

Let $D$ be the set of $x\in X$ at which $(f_{m_{k}}(x))_{k}$ is convergent; $D\in\mathcal{F}$ by Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps (iii), and $X\setminus Z\subseteq D$. Define $f:X\to\mathbb{R}$ by $f(x)=\lim_{k}f_{m_{k}}(x)$ for $x\in D$, the limit, and $f(x)=0$ for $x\notin D$. Then $f_{m_{k}}\mathbf{1}_{D}\to f$ at every point (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (i) off $D$), and the functions $f_{m_{k}}\mathbf{1}_{D}$ are measurable (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i)), so $f$ is measurable by Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps (ii). For $x\in X\setminus Z$ we have $f_{m_{k}}(x)\to f(x)$ and $|f_{m_{k}}(x)|\le g(x)$ for every $k$, hence $|f(x)|\le g(x)$ (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (viii), Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x)). As $\mu(Z)=0$, these hold for $\mu$-almost every $x$ (Null Sets and Properties Holding Almost Everywhere (i), A Property Holding Almost Everywhere (i)), and $f\in\mathcal{L}^{p}(\mu)$ by Elementary Properties of the p-Seminorm (v). This proves (iii).

Step 4: convergence of the subsequence. For $k\in\mathbb{N}$ let $u_{k}=|f_{m_{k}}-f|^{p}$, a measurable function with nonnegative values (Minkowski's Inequality (i), Power-Integrable Functions and the p-Seminorm (i)). Let $x\in X\setminus Z$. Then $|f_{m_{k}}(x)-f(x)|\to0$ (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi), Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (viii)); given $\varepsilon>0$, the number $\varepsilon^{1/p}$ is positive, and $|f_{m_{k}}(x)-f(x)|<\varepsilon^{1/p}$ gives $0\le u_{k}(x)<\varepsilon$ (Properties of Real Powers (vi), Properties of Real Powers (vii)), so $u_{k}(x)\to0$. Moreover, by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (xi),

$$|f_{m_{k}}(x)-f(x)|\le|f_{m_{k}}(x)|+|f(x)|\le2g(x),\qquad\text{so}\qquad u_{k}(x)\le|2g(x)|^{p}$$

(Properties of Real Powers (vi)). Here $2g\in\mathcal{L}^{p}(\mu)$ by Elementary Properties of the p-Seminorm (iii), so $|2g|^{p}\in\mathcal{L}^{1}(\mu)$ by Elementary Properties of the p-Seminorm (ii). By The Lebesgue Integral and Null Sets (viii), applied to $(u_{k})$, the limit $0$ and the dominating function $|2g|^{p}$, and by Elementary Properties of the p-Seminorm (ii),

$$\|f_{m_{k}}-f\|_{p}^{p}=\int u_{k}\,d\mu\to0 .$$

Given $\varepsilon>0$, there is therefore $K$ with $\|f_{m_{k}}-f\|_{p}^{p}<\varepsilon^{p}$, hence $\|f_{m_{k}}-f\|_{p}<\varepsilon$ (Properties of Real Powers (vi)), for $k\ge K$; that is, $\|f_{m_{k}}-f\|_{p}\to0$.

Step 5: convergence of the whole sequence. Let $\varepsilon>0$. First take $N$ with $\|f_{m}-f_{l}\|_{p}<\varepsilon/2$ for all $m,l\ge N$, then $K$ with $\|f_{m_{k}}-f\|_{p}<\varepsilon/2$ for $k\ge K$, and put $k=N+K$; then $k\ge K$ and $m_{k}\ge k\ge N$ (Monotone Sequences and Subsequences: Comparison of All Terms, Growth of the Indices, and Subsequences of Subsequences (ii), Arithmetic and Order of the Natural Numbers (ix)). For every $m\ge N$, since $f_{m}-f=(f_{m}-f_{m_{k}})+(f_{m_{k}}-f)$, Minkowski's Inequality (ii) gives

$$\|f_{m}-f\|_{p}\le\|f_{m}-f_{m_{k}}\|_{p}+\|f_{m_{k}}-f\|_{p}<\varepsilon$$

(Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vii)). Hence $\|f_{m}-f\|_{p}\to0$ (Convergent Sequences of Real Numbers (i)), which is (ii).

Completeness. Assume the axiom of dependent choice, write $V$ for the real normed space $(L^{p}(\mu),\|\cdot\|_{p})$ of (i), and let $(F_{m})$ be a Cauchy sequence for its norm metric $d_{V}$. Each $F_{m}$ is a subset of $\mathcal{L}^{p}(\mu)$ with an element (Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (v), Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (iv)), so countable choice, which follows from dependent choice by Countable Choice Follows from Dependent Choice (i), gives a sequence $(f_{m})$ in $\mathcal{L}^{p}(\mu)$ with $f_{m}\in F_{m}$, that is $F_{m}=[f_{m}]$, for every $m$; this is the only use of choice. By (i),

$$d_{V}(F_{m},F_{l})=\|[f_{m}]-[f_{l}]\|_{p}=\|f_{m}-f_{l}\|_{p},$$

so $(f_{m})$ is Cauchy for the $p$-seminorm, and (ii) gives $f\in\mathcal{L}^{p}(\mu)$ with $d_{V}(F_{m},[f])=\|f_{m}-f\|_{p}\to0$. Thus $(F_{m})$ converges to $[f]$ (Convergent Sequence in a Metric Space (i)), the metric space $(L^{p}(\mu),d_{V})$ is complete (Complete Metric Space (i)), and $(L^{p}(\mu),\|\cdot\|_{p})$ is a real Banach space. This proves (iv).