Lebesgue integral of Gaussian process is Gaussian.

Let $X:\Omega\times\mathcal{T}\rightarrow\mathbb{R}$ be a measurable stochastic process, Lebesgue integrable on $\mathcal{T}$ for a.e. $\omega\in\Omega$: $$\int_{\mathcal{T}} |X(\omega,t)|\,dt<\...