Suppose $f(x)$ and $g(x)$ are two bounded nonnegative functions, does it hold that if $g(x)\leq L$ then $$\int_0^T f(x)g(x)dx\leq L \int_0^T f(x)dx$$
I think the answer is yes as it follows from monotonicity property of Lebesgue integral and $f(x)g(x)\leq f(x)L$. Since $f(x)$ and $g(x)$ are bounded, the inequality above indeed make sense. Is it right?