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? 



Expectation of time integral of Wiener process
For approach 2, Fubini's theorem works just as well if you show
$$ \int\_0^T E|W\_s|ds < \infty $$
which is easy. Indeed, perhaps even easier is to note
$$ \int\_0^T E(|W\_s|^2)ds = \int\_0^T s ds = \frac{1}{2}T^2 < \infty$$
and use Jensen/Hölder/Cauchy-Schwarz.


