Can we write $\int f(x)g(x)dx \leq C\int g(x)dx$ given that $\sup_x|f(x)|\leq C<+\infty$?

Question

Can we write $\int f(x)g(x)dx \leq C\int g(x)dx$ given that $\sup_x|f(x)|\leq C<+\infty$? My answer is yes, but I'm confused with the following $\int f(x)dx \leq C\int dx$. What $\int dx$ means, is it just a length of integration, i.e. $\int_a^bdx = b -a$?

Answer 1

The expression $\int f(x)g(x)dx$ is in general a function (or strictly speaking, a set of functions). It is not a real number and thus it does not make sense to say that $\int f(x)g(x)dx\leq C\int g(x)dx$ as two sets of functions are generally non-comparable. You may consider rewriting your question using definite integrals.

Answer 2

You are correct on both counts.

For your first question, you are using the fact that $p(x) \leq q(x) \Rightarrow \int_a^bp(x)dx \leq \int_a^bq(x)dx$. [As commented by PhoemueX, we need $g(x) \geq 0$, otherwise we cannot conclude $f(x)g(x) \leq Cg(x)$ from $f(x) \leq C$]

Your second question is just an application of the first where $g(x) = 1$.