I have expectation of a random variable on a probability space $(\Omega,\Sigma,P)$ defined as: If X is lebesgue integrable with respect to $P$ then $EX = \int_\Omega X \ dP$. What I don't understand is what the author means by "lebesgue integrable with respect to $P$". To my understanding, $X$ being lebesgue integrable means that the integral of $X$ with respect to the lebesgue measure exists, but I don't know what Lebesgue integrable with respect to $P$ means. Does it mean it is both lebesgue integrable and integrable with respect to the measure $P$?
Could someone offer some intuition on this please