Let $B$ be an unbounded closed operator on a Hilbert space $H$. If $B=\int \lambda d E_\lambda $ is positive self-adjoint and a positive bounded operator $X$ commutes with every $E_\lambda $, then why $BX$ is positive and self-adjoint?

I am struggling in dealing with unbounded operators...

see page 48, line +6 (just consider $p=1$) in link. I want to understand from line 5 to line 8.

I know it is symmetric, but I have no idea why it is self-adjoint.