= Mean-square fundamental theorem of calculus
{title2=$X(t)-X(s)=\int_s^tY(u)\,du\quad\text{in }L^2$}
If an <L2 space>-valued map $X$ has a continuous mean-square derivative $Y$, then its increments equal the <Bochner integral> of $Y$. To see this, for each partition interval $[t_j,t_{j+1}]$, continuous linear observations and the scalar mean-value bound give $\|X(t_{j+1})-X(t_j)-(t_{j+1}-t_j)Y(t_j)\|_2\leq(t_{j+1}-t_j)\sup_{u\in[t_j,t_{j+1}]}\|Y(u)-Y(t_j)\|_2$. Sum these errors. Uniform continuity of $Y$ makes the total error tend to zero as the mesh tends to zero, while the <Riemann sums> converge to the <Bochner integral>. This proves the <fundamental theorem of calculus> in <L2 space>. If a <Gaussian process> derivative also has a <continuous modification>, its anchored path integral $X(s)+\int_s^t\widetilde Y(u)\,du$ therefore produces a differentiable <modification of a stochastic process> of the original <Gaussian process>.
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