# Ngô Quốc Anh

## November 10, 2009

### A trivial identity of probability measures

Filed under: Các Bài Tập Nhỏ, Giải Tích 6 (MA5205) — Ngô Quốc Anh @ 14:36

Let us consider a probability space $(X,\mathcal B,\mu)$, i.e., $(X,\mathcal B,\mu)$ is a measurable space together with $\mu(X)=1$. We assume further that $A, B \in \mathcal B$ are such that $\mu(A)=\mu(B)=1$. Then we conclude that $\mu(A \cap B)=1$.

Indeed, since $A \subset A \cup B \subset X$ then $1=\mu(A\cup B)$. We write $A \cup B$ in the following way

$A\cup B = A\backslash B \quad \bigcup \quad A \cap B \quad\bigcup \quad B\backslash A$.

We then see that $\mu(A\backslash B)=0$ since $A\backslash B \subset X\backslash B$. Similarly, $\mu(B\backslash A)=0$. Hence, $\mu(A \cap B)=1$.