Let $x(t)$ be some random path $t\in[a,b]\subset\mathbb{R}$. I.e. $x:\Omega\rightarrow\mathbb{R}^{[a,b]}$ etc.
When is $\int_a^b x(t)dt$ defined?
If $x(t)$ is Brownian motion, I know it's ok.
A much simpler example is $x(0)=1$, waits an exponential length of time $T$ and then jumps down to zero. Thus $\int_0^\infty x(t)dt$ has the same distribution as $T$.
I think the question this: Under what conditions is a stochastic process Lebesgue integrable on some interval?
Is it just that the pre-image of measurable sets in $\mathbb{R}$ are measurable w.r.t. the product measure on $\Omega\times[a,b]$? How does this translate to properties of the stochastic process, e.g. bounded variation, etc.? Any hints or references are appreciated!