Real and Complex analysis



  1. A collection M\mathfrak M of subsets of a set XX is said to be a σ\sigma-algebra in XX if M\mathfrak M has following properties:
    1. XMX \in \mathfrak M
    2. If AMA \in \mathfrak M, then AcMA^c \in \mathfrak M, where AcA^c is the complement of AA relative to XX.
    3. If An=1AnA \in \bigcup_{n=1}^{\infty} A_n and if AnMA_n \in \mathfrak M for nn =1, 2, 3, ..., then AMA\in\mathfrak M.
  2. If M\mathfrak M is σ\sigma-algebra in XX, the XX is a measurable space. and the members of M\mathfrak M are called the measurable sets in XX.
  3. If XX is a measurable space, YY is topological space, and f:XYf:X \to Y, the ff is said to be measurable provided that f1(v)f^{-1}(v) is a measurable set in XX for every open set vv in YY.