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

Each result cited is universally quantified over the data appearing in its own statement, and is applied here to the data named in the statement above. We use countable choice, to select representatives and indices indexed by $\mathbb{N}$.

Throughout, $\mathcal{L}^{p}$ is a real vector space with the pointwise operations, by Minkowski's Inequality (ii), and for $f\in\mathcal{L}^{p}$ the class $[f]$ is as in The Lebesgue Space Lᵖ(μ) (i).

Claim 1. The two operations on $L^{p}$ are well defined by The Lebesgue Space Lᵖ(μ) (ii). Each of the conditions 1 to 8 of Vector Space over a Field is an identity between classes whose two sides are, by the definition of the operations, the classes of the two sides of the corresponding identity in $\mathcal{L}^{p}$; since those identities hold in $\mathcal{L}^{p}$, the conditions hold in $L^{p}$. For instance $([f]+[g])+[h]=[(f+g)+h]=[f+(g+h)]=[f]+([g]+[h])$, and $[f]+[0_{X}]=[f+0_{X}]=[f]$, where $0_{X}$ is the map taking the value $0$ everywhere, so $[0_{X}]$ is a zero vector and is the zero vector by claim 1 of Elementary Identities in a Vector Space; and $[f]+[-f]=[f+(-f)]=[0_{X}]$ gives condition 4. So $L^{p}$ is a vector space over $\mathbb{R}$.

The number $\lVert[f]\rVert_{p}$ is well defined and nonnegative by The Lebesgue Space Lᵖ(μ) (iii). If $\lVert[f]\rVert_{p}=0$ then $f=0$ almost everywhere by Elementary Properties of the p-Seminorm (v), that is $f\sim 0_{X}$, so $[f]=[0_{X}]$ is the zero vector; this is condition (a) of Norms and Real Normed Spaces (i). Condition (b) holds because $\lVert c[f]\rVert_{p}=\lVert[cf]\rVert_{p}=\lVert cf\rVert_{p}=|c|\,\lVert f\rVert_{p}$ by Elementary Properties of the p-Seminorm (ii), and condition (c) because $\lVert[f]+[g]\rVert_{p}=\lVert f+g\rVert_{p}\le\lVert f\rVert_{p}+\lVert g\rVert_{p}$ by Minkowski's Inequality (i). Hence $L^{p}$ is a real normed space. Its distance is $d_{p}([f],[g])=\lVert[f]-[g]\rVert_{p}$ by Norms and Real Normed Spaces (iii), and $[f]-[g]=[f]+[-g]=[f-g]$, which gives the stated formula.

A construction. Let $(F_{m})_{m\in\mathbb{N}}$ be a Cauchy sequence in $(L^{p},d_{p})$ and for each $m$ let $f_{m}\in\mathcal{L}^{p}$ be a representative of $F_{m}$. We construct natural numbers $m_{1}<m_{2}<\dots$, a function $h\in\mathcal{L}^{p}$, a null set $N$ and a function $g\in\mathcal{L}^{p}$ such that for every $x\in X\setminus N$ the sequence $(f_{m_{j}}(x))_{j}$ converges to $g(x)$ with $|f_{m_{j}}(x)|\le h(x)$ for every $j$, and such that $\lVert f_{m_{j}}-g\rVert_{p}$ converges to $0$.

Step 1. For each $j\in\mathbb{N}$ the Cauchy condition, applied with the positive number $(1/2)^{j}$, provides $M_{j}\in\mathbb{N}$ with $d_{p}(F_{k},F_{l})<(1/2)^{j}$ whenever $k,l\ge M_{j}$; choose such an $M_{j}$ for each $j$. Define $m_{1}=M_{1}$ and, recursively, $m_{j+1}$ to be the larger of $m_{j}+1$ and $M_{j+1}$. Then $m_{j}<m_{j+1}$ for every $j$, and $m_{j}\ge M_{j}$, so both $m_{j}$ and $m_{j+1}$ are at least $M_{j}$ and

$$\lVert f_{m_{j+1}}-f_{m_{j}}\rVert_{p}=d_{p}(F_{m_{j+1}},F_{m_{j}})<\Bigl(\frac{1}{2}\Bigr)^{j},$$

the first equality by claim 1. Write $u_{j}=f_{m_{j}}$.

Step 2. For $J\in\mathbb{N}$ put $T_{J}=\sum_{j=1}^{J}|u_{j+1}-u_{j}|$, a finite sum in $\mathcal{L}^{p}$, and $G_{J}=|u_{1}|+T_{J}$; also put $T_{0}=0_{X}$ and $G_{0}=|u_{1}|$. Each $|u_{j+1}-u_{j}|$ lies in $\mathcal{L}^{p}$ with the same seminorm as $u_{j+1}-u_{j}$, by Elementary Properties of the p-Seminorm (iii) applied in both directions, and likewise for $|u_{1}|$. By Minkowski's Inequality (iii),

$$\lVert G_{J}\rVert_{p}\le\lVert u_{1}\rVert_{p}+\sum_{j=1}^{J}\lVert u_{j+1}-u_{j}\rVert_{p}\le\lVert u_{1}\rVert_{p}+\sum_{j=1}^{J}\Bigl(\frac{1}{2}\Bigr)^{j}\le\lVert u_{1}\rVert_{p}+1=:C,$$

the last inequality because the partial sums of the series of the $(1/2)^{j}$ are at most its sum, which is $1$, by claims 2 and 4 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series. By Elementary Properties of the p-Seminorm (i) and Properties of Real Powers (v) this gives $\int_{X}G_{J}^{p}\,d\mu=(\lVert G_{J}\rVert_{p})^{p}\le C^{p}$ for every $J$.

The sequence $(G_{J})_{J}$ is pointwise nondecreasing, since each added term is nonnegative, so $(G_{J}^{p})_{J}$ is pointwise nondecreasing by Properties of Real Powers (v). Let $H:X\to[0,\infty]$ be its pointwise least upper bound. By Monotone Convergence Theorem the map $H$ is measurable and $\int_{X}H\,d\mu$ is the least upper bound of the numbers $\int_{X}G_{J}^{p}\,d\mu$, hence at most $C^{p}$ and in particular finite. By The Lebesgue Integral and Null Sets (v) the set $N=\{x\in X:H(x)=\infty\}$ belongs to $\mathcal{F}$ and has measure $0$; it is therefore null.

Step 3. Let $\tilde H:X\to\mathbb{R}$ be the map equal to $H$ off $N$ and to $0$ on $N$; it takes real values because $H$ is finite off $N$, and it is nonnegative. It is measurable: for a real $c<0$ the set $\{x\in X:\tilde H(x)>c\}$ is $X$, while for $0\le c$ it equals $\{x\in X:H(x)>c\}\cap(X\setminus N)$, a member of $\mathcal{F}$ because $H$ is measurable and $N\in\mathcal{F}$; the criterion in Measurable Functions and the Lebesgue Integral: Standing Notation (iii) applies. Put $h=|\tilde H|^{1/p}$, which is measurable by Elementary Properties of the p-Seminorm (vi) applied with the exponents $1/p$ and $p$, and satisfies $h^{p}=\tilde H\le H$ pointwise by Properties of Real Powers (iv). Hence $\int_{X}h^{p}\,d\mu\le\int_{X}H\,d\mu<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, so $h\in\mathcal{L}^{p}$.

Let $x\in X\setminus N$, so that $H(x)=\tilde H(x)$ is a nonnegative real number. For every $J$ we have $(G_{J}(x))^{p}\le H(x)$, hence $G_{J}(x)\le h(x)$ by Properties of Real Powers (vi). Since $|u_{j}(x)|\le G_{j-1}(x)$ for every $j$, by the triangle inequality of claim 5 of Properties of the Absolute Value in an Ordered Field applied along the telescoping identity $u_{j}=u_{1}+\sum_{i=1}^{j-1}(u_{i+1}-u_{i})$, we obtain

$$|u_{j}(x)|\le h(x)\qquad\text{for every }j\in\mathbb{N}.$$

Moreover $(T_{J}(x))_{J}$ is nondecreasing and bounded above by $h(x)$, so it converges by A Bounded Monotone Sequence of Real Numbers Converges (i) and is therefore a Cauchy sequence of real numbers. For $l>j\ge 1$ the same triangle inequality gives

$$|u_{l}(x)-u_{j}(x)|\le\sum_{i=j}^{l-1}|u_{i+1}(x)-u_{i}(x)|=T_{l-1}(x)-T_{j-1}(x),$$

so $(u_{j}(x))_{j}$ is a Cauchy sequence of real numbers and hence converges, by The Real Numbers: Standing Notation and Background (iii).

Step 4. Define $g:X\to\mathbb{R}$ by letting $g(x)$ be the limit of $(u_{j}(x))_{j}$ for $x\notin N$ and $g(x)=0$ for $x\in N$. Writing $\tilde u_{j}=u_{j}\,\mathbf{1}_{X\setminus N}$ for the map equal to $u_{j}$ off $N$ and to $0$ on $N$, which is measurable by claims 1 and 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps since $X\setminus N\in\mathcal{F}$, the sequence $(\tilde u_{j}(x))_{j}$ converges to $g(x)$ at every $x\in X$, so $g$ is measurable by claim 5 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps. Passing to the limit in the bound $|u_{j}(x)|\le h(x)$, valid off $N$, and using $g=0=|g|\le h$ on $N$ together with $h\ge0$, we get $|g|\le h$ pointwise, so $g\in\mathcal{L}^{p}$ by Elementary Properties of the p-Seminorm (iii).

For $x\notin N$ the sequence $(|u_{j}(x)-g(x)|)_{j}$ converges to $0$, hence so does $((|u_{j}(x)-g(x)|)^{p})_{j}$ by Properties of Real Powers (vii) and $0^{p}=0$. Also $|u_{j}-g|\le|u_{j}|+|g|\le 2h$ pointwise off $N$, by claims 2 and 5 of Properties of the Absolute Value in an Ordered Field, so $|u_{j}-g|^{p}\le 2^{p}h^{p}$ there, by Properties of Real Powers (v) and Properties of Real Powers (iii); the map $2^{p}h^{p}$ is integrable since $h\in\mathcal{L}^{p}$. The maps $|u_{j}-g|^{p}$ are measurable by Power-Integrable Functions and the p-Seminorm (i). Applying The Lebesgue Integral and Null Sets (vii), with the constant limit function $0_{X}$, gives

$$\lim_{j\to\infty}\int_{X}|u_{j}-g|^{p}\,d\mu=0 .$$

By Elementary Properties of the p-Seminorm (i) the integral equals $(\lVert u_{j}-g\rVert_{p})^{p}$, and applying Properties of Real Powers (vii) with the exponent $1/p$ to this sequence of nonnegative reals, together with $((\lVert u_{j}-g\rVert_{p})^{p})^{1/p}=\lVert u_{j}-g\rVert_{p}$ and $0^{1/p}=0$, we conclude that $\lVert u_{j}-g\rVert_{p}$ converges to $0$. This completes the construction.

Claim 2. Let $(F_{m})$ be a Cauchy sequence in $(L^{p},d_{p})$; choose a representative $f_{m}$ of each $F_{m}$ and run the construction, obtaining $m_{j}$, $g$ and the convergence $\lVert f_{m_{j}}-g\rVert_{p}\to0$, that is $d_{p}(F_{m_{j}},[g])\to 0$.

Let $\varepsilon$ be a positive real number. There is $M\in\mathbb{N}$ with $d_{p}(F_{k},F_{l})<\varepsilon/2$ for all $k,l\ge M$, and there is $j$ with $m_{j}\ge M$ and $d_{p}(F_{m_{j}},[g])<\varepsilon/2$, because $m_{j}\ge j$ for every $j$ and $d_{p}(F_{m_{j}},[g])\to0$. For every $k\ge M$ the triangle inequality in the metric $d_{p}$ gives

$$d_{p}(F_{k},[g])\le d_{p}(F_{k},F_{m_{j}})+d_{p}(F_{m_{j}},[g])<\varepsilon .$$

Hence $(F_{m})$ converges to $[g]\in L^{p}$. So every Cauchy sequence in $(L^{p},d_{p})$ converges, that is $(L^{p},d_{p})$ is complete, and $L^{p}$ is a real Banach space by Norms and Real Normed Spaces (v).

Claim 3. Let $(F_{m})$ converge to $F$, with representatives $f_{m}$ and $f$ as in the statement. The sequence $(F_{m})$ is Cauchy: given a positive $\varepsilon$, choose $M$ with $d_{p}(F_{m},F)<\varepsilon/2$ for $m\ge M$; then $d_{p}(F_{k},F_{l})\le d_{p}(F_{k},F)+d_{p}(F,F_{l})<\varepsilon$ for $k,l\ge M$. Run the construction with these representatives, obtaining $m_{1}<m_{2}<\dots$, a function $h\in\mathcal{L}^{p}$, a null set $N_{1}$ and $g\in\mathcal{L}^{p}$ with $(f_{m_{j}}(x))_{j}$ converging to $g(x)$ and $|f_{m_{j}}(x)|\le h(x)$ for every $x\notin N_{1}$ and every $j$, and with $d_{p}(F_{m_{j}},[g])\to0$.

By A Subsequence of a Convergent Sequence Has the Same Limit the subsequence $(F_{m_{j}})_{j}$ converges to $F$, so for every positive $\varepsilon$ we have, for large $j$,

$$\lVert g-f\rVert_{p}=d_{p}([g],F)\le d_{p}([g],F_{m_{j}})+d_{p}(F_{m_{j}},F)<\varepsilon .$$

Hence $\lVert g-f\rVert_{p}=0$, by Comparison of Real Numbers with Arbitrary Positive Slack, so $g-f=0$ almost everywhere by Elementary Properties of the p-Seminorm (v); let $N_{2}$ be the null set of points where $g$ and $f$ differ. By The Lebesgue Integral and Null Sets (i) the set $N=N_{1}\cup N_{2}$ is null, and for $x\in X\setminus N$ the sequence $(f_{m_{j}}(x))_{j}$ converges to $g(x)=f(x)$ and satisfies $|f_{m_{j}}(x)|\le h(x)$ for every $j$. This is the assertion.