Let $X$ be a real random variable with c.d.f function $F$.
Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite).
What additional assumptions do I need on $g$ for the following equality to hold?
$$ \mathbb{E}\left[g(X)\right] = – \int_{-\infty}^{0}{F(t) \ dg(t)} + g(0) + \int_{0}^{+\infty}{\left(1-F(t) \right) \ dg(t)} $$
I have seen people using these kind of equalities, but I have never seen a rigorous statement yet. So I would like to know when can I use this transformation, and furthermore I am looking for a reference I can cite when using it.
Thank you for your help.
Edit: Equality corrected thanks to Alexandre Eremenko's comments.
Best Answer
The formula you wrote is incorrect: if you add a constant to $g$, the left hand side will change while the right hand side will not. The correct integration by parts formula is $$\int_{-\infty}^\infty gdF=-\int_{-\infty}^0Fdg+g(0)+\int_0^\infty(1-F)dg.$$ You need some condition at $\pm\infty$ that guarantees that $gF\to 0$ as $t\to-\infty$, and $g(1-F)\to 0$ as $t\to+\infty$. And of course that the functions do not jump at $0$.