Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge0}$ be a complete filtration of $\mathcal A$
- $H$ be a $\mathbb R$-Hilbert space
- $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
- $X$ be an $L^2$-bounded almost surely continuous $H$-valued $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ with $X_0=0$ almost surely and $$X^n:=\langle X,e_n\rangle_H\;\;\;\text{for }n\in\mathbb N$$
- $[X^n]$ denote the quadratic variation of $X^n$ for $n\in\mathbb N$
Note that $$S:=\sup_{N\in\mathbb N}\sum_{n=1}^N[X^n]\xleftarrow{N\to\infty}\sum_{n=1}^N[X^n]\tag1$$ is $\mathcal F\otimes\mathcal B([0,\infty))$-measurable. Since $$\operatorname E\left[[X^n]_t\right]=\operatorname E\left[|X_t^n|^2\right]<\infty\tag2\;,$$ we obtain $$\operatorname E[S_t]\xleftarrow{N\to\infty}\operatorname E\left[\sum_{n=1}^N[X^n]_t\right]=\operatorname E\left[\sum_{n=1}^N|\langle X_t,e_n\rangle_H|^2\right]\xrightarrow{N\to\infty}\operatorname E\left[\left\|X_t\right\|_H^2\right]\tag3$$ and hence $$S_t<\infty\;\;\;\text{almost surely}\tag4$$ for all $t\ge0$.
However, what I need is $$S_t<\infty\;\;\;\text{for all }t\ge0\text{ almost surely}\tag5\;,$$ i.e. a common $\operatorname P$-null set over all $t\ge0$ in $(4)$. How are we able to show that?