Proof of Separability of the Lebesgue Space of Square-Integrable Vector-Valued Functions

Write $\lambda=\lambda_{[0,T]}$, $\mathcal{B}=\mathcal{B}_{[0,T]}$, $P_{T}=T^{-1}\lambda$, and let $\mathcal{G}_{m}$, $I_{m,p}$, $A_{m}$ be as in the dyadic averaging lemma.

Claim 1. Square-integrability. Let $w=\sum_{p=1}^{2^{m}}c_{p}\mathbf{1}_{I_{m,p}}\in D_{m}$. Its $i$-th component is $w^{i}=\sum_{p}c_{p}^{i}\mathbf{1}_{I_{m,p}}$, a simple function built from the sets $I_{m,p}\in\mathcal{B}$, hence $\mathcal{B}$-measurable. Since the atoms are pairwise disjoint with union $[0,T]$, $w$ takes only the values $c_{1},\dots,c_{2^{m}}$, so with $C$ the greatest element of the finite family $\bigl(|c_{p}|\bigr)_{p\in\{1,\dots,2^{m}\}}$, which exists by Greatest Element of a Finite Family in a Totally Ordered Set, we get $|w|^{2}\le C^{2}$ pointwise and, by monotonicity, $\int_{[0,T]}|w|^{2}\,d\lambda\le C^{2}\lambda([0,T])=C^{2}T<\infty$. So $w\in\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$.

Countability. Fix $m\in\mathbb{N}$. The set $\mathbb{Q}^{d}$ is countable by claim 3 of The Integers and the Rational Numbers are Countable, hence so is the set $(\mathbb{Q}^{d})^{2^{m}}$ of $2^{m}$-tuples in $\mathbb{Q}^{d}$, by claim 2 of Products and Powers of Countable Sets. The map sending a tuple $(c_{1},\dots,c_{2^{m}})$ to $\sum_{p=1}^{2^{m}}c_{p}\mathbf{1}_{I_{m,p}}$ has image exactly $D_{m}$, so $D_{m}$ is countable by claim 4 of Basic Properties of Countable Sets. The sets $D_{m}$ form a family of subsets of the set of all maps $[0,T]\to\mathbb{R}^{d}$, indexed by $\mathbb{N}$, so their union $D$ is countable by A Countable Union of Countable Sets is Countable.

Claim 2. Let $u\in\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ and let $\varepsilon>0$ be real.

Step A: dyadic averages approximate $u$. By claim 7 of the inner-product lemma, each component $u^{i}$ is a square-integrable random variable on $([0,T],\mathcal{B},P_{T})$. By claim 3 of the dyadic averaging lemma, $A_{m}u^{i}$ is a conditional expectation of $u^{i}$ given $\mathcal{G}_{m}$; by claim 1 of that lemma the $\mathcal{G}_{m}$ ($m\ge1$) form a nondecreasing sequence of sub-$\sigma$-algebras of $\mathcal{B}$; and by claim 2 of that lemma their generated $\sigma$-algebra is $\mathcal{B}$ itself. Since $u^{i}$ is $\mathcal{B}$-measurable and square-integrable, the choice $Y=u^{i}$ satisfies conditions (i), (ii), (iii) of the definition of conditional expectation with $\mathcal{G}=\mathcal{B}$, so $u^{i}$ is a conditional expectation of $u^{i}$ given $\mathcal{B}$. Levy's upward theorem in mean square, applied to $u^{i}$ and this filtration, therefore gives $\lVert A_{m}u^{i}-u^{i}\rVert_{2}\to0$ as $m\to\infty$.

Let $A_{m}u:[0,T]\to\mathbb{R}^{d}$ be the map with components $(A_{m}u)^{i}=A_{m}u^{i}$, that is

$$A_{m}u=\sum_{p=1}^{2^{m}}b_{m,p}\,\mathbf{1}_{I_{m,p}},\qquad b_{m,p}=2^{m}\Bigl(\mathbb{E}\bigl[u^{1}\mathbf{1}_{I_{m,p}}\bigr],\dots,\mathbb{E}\bigl[u^{d}\mathbf{1}_{I_{m,p}}\bigr]\Bigr)\in\mathbb{R}^{d}.$$

Each $A_{m}u^{i}$ is bounded and $\mathcal{B}$-measurable by claim 3 of the dyadic averaging lemma, so $A_{m}u\in\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ by the argument already given in claim 1. By claim 7 of the inner-product lemma,

$$\lVert[A_{m}u]-[u]\rVert_{L^{2}}^{2}=T\sum_{i=1}^{d}\lVert A_{m}u^{i}-u^{i}\rVert_{2}^{2},$$

a finite sum of real sequences each with limit $0$, hence with limit $0$. Fix $m\ge1$ with $\lVert[A_{m}u]-[u]\rVert_{L^{2}}<\varepsilon/2$ and abbreviate $b_{p}=b_{m,p}$.

Step B: rounding the values to rational vectors. Put $\eta=\varepsilon\bigl(2\sqrt{Td}\bigr)^{-1}$, a positive real number. By claim 2 of The Rational Numbers are Dense in the Real Numbers there is, for each $p\in\{1,\dots,2^{m}\}$ and each $i\in\{1,\dots,d\}$, a rational number $c_{p}^{i}$ with $\bigl|b_{p}^{i}-c_{p}^{i}\bigr|<\eta$. The vector $c_{p}=(c_{p}^{1},\dots,c_{p}^{d})$ then lies in $\mathbb{Q}^{d}$, and by claim 1 of the elementary properties of the Euclidean norm,

$$|b_{p}-c_{p}|^{2}=\sum_{i=1}^{d}\bigl(b_{p}^{i}-c_{p}^{i}\bigr)^{2}<d\,\eta^{2}=\frac{\varepsilon^{2}}{4T}.$$

Define

$$w=\sum_{p=1}^{2^{m}}c_{p}\,\mathbf{1}_{I_{m,p}},$$

which lies in $D_{m}$ and hence in $D$. Let $\kappa$ be the greatest element of the finite family $\bigl(|b_{p}-c_{p}|^{2}\bigr)_{p\in\{1,\dots,2^{m}\}}$, which exists by Greatest Element of a Finite Family in a Totally Ordered Set; then $\kappa<\varepsilon^{2}(4T)^{-1}$. By claim 1 of the dyadic averaging lemma the level-$m$ atoms are pairwise disjoint with union $[0,T]$, so each $t\in[0,T]$ lies in exactly one atom $I_{m,p}$, and there $A_{m}u$ takes the value $b_{p}$ and $w$ the value $c_{p}$, whence $|w(t)-A_{m}u(t)|^{2}=|c_{p}-b_{p}|^{2}\le\kappa$. Therefore, by monotonicity,

$$\lVert[w]-[A_{m}u]\rVert_{L^{2}}^{2}=\int_{[0,T]}|w-A_{m}u|^{2}\,d\lambda\le\kappa\,T<\frac{\varepsilon^{2}}{4},$$

and since both sides are nonnegative, $\lVert[w]-[A_{m}u]\rVert_{L^{2}}<\varepsilon/2$.

Conclusion. Note first that $\lVert v-v'\rVert_{L^{2}}=\lVert v'-v\rVert_{L^{2}}$ for all $v,v'$, by the absolute homogeneity in claim 5 of the inner-product lemma applied to the scalar $-1$. By the triangle inequality of that same claim,

$$\lVert[u]-[w]\rVert_{L^{2}}\le\lVert[u]-[A_{m}u]\rVert_{L^{2}}+\lVert[A_{m}u]-[w]\rVert_{L^{2}}<\varepsilon .$$

Density. Let $U$ be a nonempty open subset of $L^{2}([0,T];\mathbb{R}^{d})$ and let $\xi\in U$. There is a real $\varepsilon>0$ with $\{\zeta:d_{L^{2}}(\zeta,\xi)<\varepsilon\}\subseteq U$. Choosing a representative $u\in\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ of $\xi$ and applying claim 2 produces $w\in D$ with $d_{L^{2}}([w],\xi)=\lVert[w]-[u]\rVert_{L^{2}}=\lVert[u]-[w]\rVert_{L^{2}}<\varepsilon$, so $[w]\in U$. Thus $\{[w]:w\in D\}$ meets every nonempty open subset, so its closure is the whole space and it is dense. Finally, $\{[w]:w\in D\}$ is the set of values of the map $w\mapsto[w]$ on $D$, so it is countable by claim 1 above and claim 4 of Basic Properties of Countable Sets.