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 and the Seminormed Space of Power-Integrable Functions §vector-space, and for $f\in\mathcal{L}^{p}$ the class $[f]$ is as in The Lebesgue Space of Power-Integrable Functions §equivalence.

Claim 1. The two operations on $L^{p}$ are well defined by The Lebesgue Space of Power-Integrable Functions §space. 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 of Power-Integrable Functions §norm. If $\lVert[f]\rVert_{p}=0$ then $f=0$ almost everywhere by Elementary Properties of the p-Seminorm §vanishing, that is $f\sim 0_{X}$, so $[f]=[0_{X}]$ is the zero vector; this is condition (a) of Real Normed Space and Real Banach Space §norm. 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 §homogeneous, 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 and the Seminormed Space of Power-Integrable Functions §minkowski. Hence $L^{p}$ is a real normed space. Its distance is $d_{p}([f],[g])=\lVert[f]-[g]\rVert_{p}$ by Real Normed Space and Real Banach Space §distance, 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

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 §comparison applied in both directions, and likewise for $|u_{1}|$. By Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions §finite-sums,

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 §power and Properties of Real Powers of Nonnegative Real Numbers §monotone 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 of Nonnegative Real Numbers §monotone. 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: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §finite 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 Measure Spaces and the Lebesgue Integral: Standing Notation §measurable applies. Put $h=|\tilde H|^{1/p}$, which is measurable by Elementary Properties of the p-Seminorm §rescaling applied with the exponents $1/p$ and $p$, and satisfies $h^{p}=\tilde H\le H$ pointwise by Properties of Real Powers of Nonnegative Real Numbers §exponents. 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, 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 of Nonnegative Real Numbers §inverse. 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

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 §nondecreasing and is therefore a Cauchy sequence of real numbers. For $l>j\ge 1$ the same triangle inequality gives

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

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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions 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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. 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 §comparison.

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 of Nonnegative Real Numbers §continuity 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 of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §product; 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 §measurable-power. Applying The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §dominated, with the constant limit function $0_{X}$, gives

By Elementary Properties of the p-Seminorm §power the integral equals $(\lVert u_{j}-g\rVert_{p})^{p}$, and applying Properties of Real Powers of Nonnegative Real Numbers §continuity 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

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 Real Normed Space and Real Banach Space §banach.

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 $j$,

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 §vanishing; let $N_{2}$ be the null set of points where $g$ and $f$ differ. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union 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.