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Firstly,

Definition 1: function f is measurable if we have a sequence of simple function $s_n$ such that $s_n \to f$.

Definition 2: a set $A$ is measurable if characteristic function $\chi_A$ is measurable.

but by these definition, I very confuse when I prove that continuous function then measurable. Can anyone give me a proof?.

PS: only use these definition.

Secondly, I have Another question: show that if $A_n$ are measurable sets then $\bigcap_{n=1}^{\infty}A_n$ also measurable set.

My solution: Beacause $A_n$ are measurable sets, characteristic functions $\chi_{A_n}$ measurable, or exist sequence of simple finction $s_n$ s.t $s_n \to \chi_{A_n}$. So $$s=\prod_{n=1}^{\infty}s_n \to \prod_{n=1}^{\infty}\chi_{A_n}=\chi_{\bigcap_{n=1}^{\infty}A_n}$$ and since $s$ also simple function, we have $\chi_{\bigcap_{n=1}^{\infty}A_n}$ measurable. Therefore $$\bigcap_{n=1}^{\infty}A_n$$ measurable. QED

I still feel something is wrong. Can anyone help me check my solution?. Thankful

  • 1
    How are simple functions defined? I think simple functions must be defined using the definition of measurable (of sets)2012-11-08
  • 0
    Here, I know this definition very confuse. But it is definition.2012-11-08

1 Answers 1