2

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and got confused at the form of integration on random variables.

In chapter 2, page 9 (sixth edition) it (consider a given probability space $(\Omega, \mathscr{F}, P)$) defines the expectation of a random variables as:

If $\inf_\Omega |X(\omega)| dP(\omega) < 1$ then the number $$E[X] := \int_\Omega X(\omega) dP(\omega) = \int_{\mathbb{R}^n}x d\mu_X(x)$$ is called the expectation of $X$ (w.r.t. P).

My questions is, why the integration form is written as $\int_\Omega X(\omega) dP(\omega)$, not $\int_\Omega X(\omega) P(\omega) d\omega$? Also $\int_{\mathbb{R}^n}x d\mu_X(x)$ looks easier to understand if written as $\int_{\mathbb{R}^n}x \mu_X(x) dx$, isn't it? At least $\int f(x) dx$ is the form I learnt in Riemann integration.

I'm wondering if there's a reason of writing the integration in that way?

athos
  • 5,177

1 Answers1

1

Re $\mu_X$, note that not every measure has a density with respect to Lebesgue measure (consider for example the uniform probability measure on the ternary Cantor set). Re $P$, note that no Lebesgue measure on $\Omega$ may exist since $\Omega$ can be basically any (non empty) set, then what would be $\mathrm d\omega$?

Probability theory is based on Lebesgue, not Riemann, integration.

Did
  • 279,727