When I was taught independence, it says that $A$ and $B$ are independent iff $P(A\cap B)=P(A)P(B)$. But now I am learning BASU's theorem, in which proof, it says that $A$ and $B$ are independent because $P\big\{P(A\cap B)=P(A)P(B)\big\}=1$.
Why does it true? Does it has any relation with measure theory? I do not know anything about measure theory.
The proof of BASU's theorem can be found at the beginning of this link. And what I ask is on the top of page 3.
In the proof of BASU's theorem, it says that $T$ is a complete statistic and that $\mathbb{E}[P(S=s|T=t)-P(S=s)]=0$. Thus $P\big(P(S=s|T=t)-P(S=s)=0 \big)=1$. Hence $S$ and $T$ are independent.