Fubini's theorem for Stochastic Integral, with sum

Question

I am struggling here with part (2), .

In usual instances, I've had the question phrased like this

but I'm not sure how to deal with the summation?

Answer 1

Hints:

1. Show that for any random variables $Y_1,\ldots,Y_n$$$\left( \sum_{k=1}^n Y_k 1_{(k/n,(k+1)/n]}(t) \right)^2 = \sum_{k=1}^n Y_k^2 1_{(k/n,(k+1)/n]}(t) \tag{1}$$ using the fact that the intervals $(k/n,(k+1)/n]$ are pairwise disjoint. 2. Applying $(1)$ and Tonelli's theorem yields $$\mathbb{E} \left( \int_0^{\infty} X_n(t)^2 \, dt \right) = \sum_{k=0}^{n-1} \int_{k/n}^{(k+1)/n} \mathbb{E}(W(k/n)^2) \, dt. \tag{2}$$
2. Use $W(t) \sim N(0,t)$ to conclude that the right-hand side of $(2)$ is finite.