1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    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!

Homework Help: Sequence of measurable functions and limit

  1. May 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Given a [tex]\sigma[/tex]-algebra [tex](X,\mathcal{A})[/tex], let [tex]f_n : X \to
    [-\infty,\infty][/tex] be a sequence of measurable functions. Prove that
    the set [tex]\{ x \in X | \lim f_n (x) \text{ exists} \}[/tex] is in

    2. Relevant equations

    Let [tex](X,\mathcal{A})[/tex] be a [tex]\sigma[/tex]-algebra and [tex]M_{+}(X) := [/tex] set
    of all functions [tex]f : X \to [0,\infty][/tex] which are
    [tex]\mathcal{A}[/tex]-measurable. Let [tex]f_n \in M_+(X)[/tex], then if [tex]\displaystyle \lim_{n \to \infty} f_n(x) \exists[/tex] [tex]\forall
    x \in X[/tex], then [tex]\displaystyle f := \lim_{n \to \infty} f_n \in

    Any [tex]f : X \to [-\infty,\infty][/tex] can be written [tex]f = f_+ - f_-[/tex]
    where [tex]f_+(x) := \sup \{ f(x), 0 \}[/tex], [tex]f_-(x) := - \inf \{ f(x), 0
    \}[/tex]. So [tex]f[/tex] is [tex]\mathcal{A}[/tex]-measurable iff [tex]f_+, f_- \in M_+(X)[/tex].

    3. The attempt at a solution

    Should I use the above two facts to show that there exists a measurable [tex]\displaystyle f = \lim_{n \to \infty} f_n[/tex] and hence anything [tex]f^{-1}(E) \in \mathcal{A}, E \in \mathbb{R}[/tex]?
  2. jcsd
  3. Aug 10, 2010 #2
    Here's a sketch of a proof.
    1. Define f(x) =lim f_n(x) if f_n(x) converges ,
    = 0 otherwise.
    2. Using Lebesgue's dominated convergence theorem, show that f is measurable.
    3. If a measurable function exists on a set S, its characteristic function ,too, is measurable.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook