MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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 $0$.

share|cite|improve this answer
Thanks a lot ! Would you know a reference for this result ? – Adrien Dec 14 '13 at 15:32
This result is called "integration by parts", any analysis textbook which has Stieltjes integral will have it. The general formula is $\int_a^bfdg=fg|_a^b-\int_a^bgdf.$ – Alexandre Eremenko Dec 14 '13 at 15:38
There is some hypothesis about points of discontinuity ...–Stieltjes_integration#Integration_by_parts – Gerald Edgar Dec 14 '13 at 15:43
Yes, but if they are not on the endpoints, it is OK. That's why I said "there is no jump at $0$. of course, the formula can be modified for the case that the points of discontinuity are at the endpoints. – Alexandre Eremenko Dec 14 '13 at 16:48
Thanks, but I'm not looking for just a reference on integration by parts for the Stieljtes on a finite interval, but for the equality mentioned. First my domain of integration is infinite. Second while obviously some kind of continuity on $g$ is required, it is not obvious to me whether the conditions on the behaviour of $gF$ and $g(1-F)$ at infinity are required when the Lebesgue integral is assumed to be finite. – Adrien Dec 14 '13 at 18:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.