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Borel measurable functions

  1. Sep 17, 2011 #1
    A function f: E -> [itex]\mathbb{R}[/itex] is called Borel measurable if for all [itex]\alpha \in \mathbb{R}[/itex] the set [itex]\{x \in R : f(x) > \alpha \}[/itex] is a Borel set.

    If f is a strictly increasing function, then f is Borel measurable.

    Proof:

    Let [itex]H=\{x \in \mathbb{R} : f(x) > \alpha \}[/itex]. I want to show that [itex](\alpha, \infty) \subset H[/itex].

    My first guess is to assume that this is non-empty or else the result is trivial.

    The next step, from a hint I was given, was to look at infimums (or supremums). However, I'm not sure how to proceed.
     
  2. jcsd
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