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.

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