= Mean-square integral
{title2=$\int_a^bX_tdt\in L^2(\Omega)$}
An ordinary time integral of <random variables> taken in the <L2 space> of the underlying <probability space>. Under strong measurability and $\int_a^b(\mathbb E|X_t|^2)^{1/2}dt<\infty$, it exists as a <Bochner integral>; for a mean-square continuous process it is also the limit of <Riemann sums> in that space. If the process is centered and the covariance is integrable, <Fubini's theorem> gives
$$
\operatorname{Var}\left(\int_a^bX_tdt\right)=\int_a^b\int_a^b\operatorname{Cov}(X_u,X_v)\,du\,dv.
$$
For a <Gaussian process>, the integral is a <Gaussian random variable>, by approximating it in mean square by finite linear combinations of jointly <Gaussian random variables>.
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