I have the following problem:
Let $H : [0, \infty] \times \Omega \rightarrow \mathbb{R}$ be adapted and jointly measurable, while assuming $\int _{0}^{t}E \mid H_{s} \mid ^{2}ds < \infty$ for all $t < \infty$ and $H_{t} = 0$ for $t < 0$.
Fix $n \geq 1$ and define $H_{n} : [0, \infty] \times \Omega \rightarrow \mathbb{R}$ by $H_{n}(t) = n \int _{t - \frac{1}{n}}^{t} H_{s} ds$. I want to prove two things:
1.) $H_{n}$ is continuous
2.) $\mid H(t) - H_{n}(t) \mid ^{2} \hspace{4pt}\leq n \int _{t - \frac{1}{n}}^{t}\mid H(t) - H(s) \mid ^{2} ds$
For continuity I just want to appeal to the continuity of the ordinary Riemann-integral, but I'm afraid I would skip some key steps by doing this. And for equation 2 I have already proven that $E[H_{n}(t)^{2}] < \infty$ for all $t \geq 0$ and that $H_{n}$ is adapted. It just seems like the integral and the square are being swapped, is there some kind of theorem that states this?