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:

1. 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.

2. 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?

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:

1. 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.

2. 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?