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Convergence in L{p} space

  1. Nov 19, 2013 #1
    If [itex]f_{n} \underset{n \to \infty}{\longrightarrow} f[/itex] in [itex]L^{p}[/itex], [itex]1 \leq p < \infty[/itex], [itex]g_{n} \underset{n \to \infty}{\longrightarrow} g[/itex] pointwise and [itex]|| g_{m} ||_{\infty} \leq M \forall n \in \mathbb{N}[/itex] prove that:
    [itex]f_{n} g_{n} \underset{n \to \infty}{\longrightarrow} fg[/itex] in [itex]L^{p}[/itex]

    My attemp:

    [itex]\displaystyle \int | f_{n} g_{n} - f g_{n} |^{p} = \displaystyle \int | g_{n} | | f_{n} - f |^{p} \leq M^{p} \displaystyle \int | f_{n} - f |^{p}[/itex]
    Then [itex]f_{n} g_{n} \underset{n \to \infty}{\longrightarrow} f g_{n}[/itex] in [itex]L^{p}[/itex]
    Now lets prove that [itex]f g_{n} \underset{n \to \infty}{\longrightarrow} fg [/itex] in [itex]L^{p}[/itex]
    [itex]g_{n} \longrightarrow g[/itex] a.e. [itex]\Longrightarrow g_{n} \underset{\longrightarrow}{m} g[/itex]
    [itex]\forall \varepsilon > 0, \forall \delta > 0, \exists n_{0} \in \mathbb{N} / \forall n \geq n_{0} :[/itex]
    [itex]| D | = | \{ x / | g_{n} (x) - g(x) | \geq \delta \} | < \varepsilon[/itex]

    [itex]\displaystyle \int | f g_{n} - fg |^{p} = \displaystyle \int | f |^{p} | g_{n} - g |^{p} \leq \displaystyle \int_{D} | f |^{p} M^{p} + \displaystyle \int_{D^{c}} | f |^{p} \delta^{p}[/itex]
    I know [itex]| D | < \varepsilon[/itex], but [itex]f[/itex] isn't necessarily bounded in [itex] D [/itex], I need to prove that [itex]\int_{D} | f |^{p} \longrightarrow 0[/itex] as [itex]| D | \to 0[/itex]

    Anyone has any idea?
    Is this approach right or the last step is false and I need to rework the proof enterely?

    Edit 2: Edit 1 was completely wrong so I deleted it.
    Last edited: Nov 19, 2013
  2. jcsd
  3. Nov 19, 2013 #2
    You'll need to redo the proof start by adding an additional term and immediately subtracting it then use your bounds and the fact that gn => g and fn=>f pointwise.
  4. Nov 19, 2013 #3
    You mean to be able to apply dominated convergence theorem?

    Anyway, I asked my teacher and he knew filled the last gap in my proof.

    [itex]\displaystyle \int | f |^{p} < \infty \Longrightarrow \displaystyle \int_{E} | f |^{p} < \infty \forall E \subset R^{n}[/itex]
    Then by the absolute continuity of the Lebesgue integral, for every [itex]\varepsilon > 0[/itex] there exists a [itex]\delta ' > 0 [/itex] such that:
    [itex]| E | < \delta 0 \Longrightarrow | \displaystyle \int_{E} | f |^{p} | < \varepsilon [/itex]
    As [itex](2M)^{p}[/itex]* is a constant, this converges to zero as [itex]\delta \to 0[/itex] and [itex]\varepsilon \to 0[/itex]
    * I made a little mistake in the previous post:
    [itex]| g - g_{n} |^{p} \leq (2M)^{p}[/itex] not only [itex]M^{p}[/itex]
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