1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Semi continuity and Borel Sets - Measurable Functions

  1. Oct 6, 2013 #1
    In a book I'm reading it says:
    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).
    Can someone give me a hint please?
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted