I have such general equation for expected value definition in this book, p 64 $$ E(X)=\int_\Omega \omega(\zeta)P(d\zeta) $$ where $\omega$ is random value and $P$ is probability function.
How to translate it to classic Riemann integral definition
$$ E(X)= \int_\Omega w(\zeta)f(\zeta)d\zeta $$
where $f(\zeta)$ is probability density? I found simular question there, but there is another designation