Mean-square integral (${\int }_a^b{X}_{t}dt\in L^2(\Omega )$)

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 givesFor 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.