Well posedness integral by parts

Question

I want to integrate by parts the following:

$$\int_{z}^{\infty} h(u) \frac{dF(u)}{\sqrt{u-z}} $$

My question would be: under which conditions the following is valid:

$$\int_{z}^{\infty} h(u) \frac{dF(u)}{\sqrt{u-z}} = \left( h(s) \int_{s}^{\infty} \frac{dF(u)}{\sqrt{u-s}}\right)\bigg|_{z}^{\infty} - \int_{z}^{\infty} \left( \int_{s}^{\infty} \frac{dF(u)}{\sqrt{u-s}} \right) dh(s) $$

Answer 1

Check here. In this answer they explain that the formula you are trying to prove is valid only if you prove that all the three integrals involved are convergent. (See the first comment by @Kavi Rama Murthy).