I am trying to calculate $E(\int_0^T {W_s ds})$, where $W_s$ is a standard Brownian motion.
Now two approaches I can think of:
Take a partition of $[0,T]$. Calculate $E(\sum {W_{t_i}(t_{i+1} - t_i)})$ and take the limit as you shrink the size of the partition.
Calculate $\int_0^T { E(W_s)ds }$.
However, for approach 1), its not clear what function would dominate the absolute value of the terms inside the E() for all possible partitions, and that it would have a finite expectation. So, interchanging limit and expectation is dicey.
For approach 2), Fubini's theorem would require me to know a-priori that $E(\int_0^T {|W_s|ds})$ is finite, and I don't see how I could show that.
How can any of these approaches be fixed, if at all? Or is there another way to solve the problem?
