Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic subsets of $X$. Can I say something general about the relation between the two following expressions?

$$\int_X f(x) g(x) dx$$ and $$\int_X f(x) dx \int_X g(x) dx$$

(for instance that the first one is always smaller or equal to the second one). If not, is there a way to prove that,

$$\frac{\int_X f(x) g(x) h(x)dx}{\int_X f(x)dx} \leq \frac{\int_X f(x) g(x) dx}{\int_X f(x)} \frac{\int_X g(x) h(x) dx}{\int_X g(x) dx},$$ where $h(x)=1$ if $h(x) \in C \subset X$ and $0$ otherwise?