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Semi continuity and Borel Sets - Measurable Functions

  1. Oct 6, 2013 #1
    In a book I'm reading it says:
    \newline
    If [itex]f: \mathbb{R} \longrightarrow \mathbb{R}[/itex] is lower semi continous, then [itex]\{f > a \}[/itex] is an open set therefore a borel set. Then all lower semi continous functions are borel functions.
    It's stated as an obvious thing but I couldn't prove it.
    The definition a lower semi continous function I'm using is:
    A function is lower semicontinous in the point [itex]x_0[/itex] if
    [itex] \underline{lim}_{x \rightarrow x_{0}} \geq f(x_{0}) [/itex]
    [itex] \underline{lim}_{x \rightarrow x_{0}}= \sup_{\delta > 0 } \{ inf \{ f(x) / |x-x_{0} | < \delta \} \} [/itex]
    (A function is lower semicontinous in [itex]\mathbb{R}^{p}[/itex] if for all [itex]x \in \mathbb{R}^{p}[/itex] it's lower semicontinous).
    [itex][/itex]
    Can someone give me a hint please?
     
  2. jcsd
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