Relationship "finite mean"  "absolutely integrable What is the relationship between the property of a random variable (i.e. a measurable function defined on some probability space) being absolutely integrable, i.e. $$\mathbb{E}|X|&lt;\infty$$and having finite mean, i.e.$$\mathbb{E}X&lt;\infty\quad ?$$Obviously the first implies the second, but does the converse hold too ?
Exact equivalence, since E(X) is only defined when E(|X|) is finite. In Lebesgue theory of integration, the integral of a real-valued measurable function $f$ exists if and only if $|f|$ has finite integral.
nope. in general it does not hold. take the sin for example..