Let $(X(t))_{t\in[0,1]}$ be a stochastic process with trajectories in $L^2[0,1]$ and continuous covariance function k(s,t). I need to show that the second absolute moment is finite, i.e. $\mathbb{E}\|X\|^2< \infty$.
This is my approach:
If $k(s,t):=\mathbb{E}[X(s)X(t)]$, then $k(t,t)=\mathbb{E}[X(t)^2]$
on the other hand, the second absolute moment is given by: $\mathbb{E}\|X\|^2=\int_{0}^{1} |X(t)|^2dP(x)$
In all places $\mathbb{E}$ denotes Bochner expectation (or strong expatation). So I am not sure how to conclude relating $k(t,t)$ and $\mathbb{E}\|X\|^2$ in the case of this expectation.
Thank you for any help.