Thank you Michele, I think I got the thing which I were looking for after discussion here. Thank you

Dear Pablo, Thank you for your wishes. All your references are in finite space. I haven't seen in general form. Anyways, its maybe correct, I need to study more.

Dear Michele, to me, it seems for your proof, we need decomposition of $\log(\frac{dQ}{dP})\frac{dQ}{dP}$ which is not concave any more... Is not it?

Dear Pablo, According to the link which you have sent, there Gibb's inequality is used and in Gibb's inequality they have used that finite point in sample space. That is my problem, if it is infinite then what is the condition.

What do you mean by partial answer? What is equal to 0 a.s. in the end?