Axioms of Expectation Proof: Probability of Arbitrary Sets

Question

If A, B are arbitrary sets, use the five axioms of expectation to prove: $P(A\cup B) = P(A)+P(B)-P(A\cap B)$.

Any hints to go about this? I can prove the case where they are discrete sets, so it's more of the case where the sets are not exclusive.

1. $X \geq 0, then E(X) \geq 0$

2. $E(cX) = cE(X)$

3. $E(X_1 + X_2) = E(X_1) + E(X_2)$

4. $E(1) = 1$

5. If a sequence of random variables $X_i(w)$ is monotonically increasing to a limit $X(w)$, then $E(X_i)$ converges to $E(X)$.

Much appreciated.

Answer 1

Hint:

Let $\raise{0.5ex}\chi$ denote the indicator variable.

prove that

$$\raise{0.5ex}\chi_{A\cup B}+\raise{0.5ex}\chi_{A\cap B}=\raise{0.5ex}\chi_{A}+\raise{0.5ex}\chi_B$$