Do we have $ \int_{A} \chi_B \mu = \mu(A) $ in a probability space?

Question

Here I have a rather naive question concerning integral representation of probability measures. In general I have problems with it, so here there is a super basic setting:

• $(X, \Sigma, \mu)$ probability space,
• $A, B \in \Sigma$,
• $\chi_B \in [0,1]^X$ indicator function of $B$ (measurable),
• $\int_X \chi_B \mu$.

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Problem:
Considering that $A \in \Sigma$ is arbitrary, can we proceed with the following equivalence: $$ \int_{A} \chi_B \mu = \mu(A) ?$$

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Questions:

I think we can in general, but I am not sure why.

1. Is it true that we can?
2. If yes, why?

Any feedback as always it is greatly appreciated.
Thank you for your time.

Answer 1

For the sake of an answer: $$ ∫_A1_Bdμ=∫_X1_A1_Bdμ=∫_X1_{A∩B}dμ=μ(A∩B). $$