I was trying to solve a probability theory exercise and wondered if the following was true:

Let $ ( Omega, mathfrak {A}, mu) $ to be a measurement space, $ A in mathfrak {A} $ and $ f in L ^ 1 ( mu) $ not negative. Then we have this $ int_ mu { chi_A f text {d} omega} leq mu (A) || f || _ {L ^ 1 ( mu)} $.

I tried to find a counterexample on the real line, but I couldn't find one. Maybe there is one really easy one that I overlooked. I tried to prove the inequality but I couldn't find the right approach. I tried to argue via the step by step functions, but it doesn't seem to work. Perhaps it is necessary to limit oneself to finite measures for the above to be true. Any help is appreciated.