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Suppose $f_n$ is a sequence of functions over measurable space $(X,\mu)$. Which one of the followings statements hold?

  1. If we have $f_n \to f$ uniformly, could we say $f_n \to f$ in measure?
  2. If we have $f_n \to f$ in $L^1(\mu)$, could we say $f_n \to f$ in measure?
  3. If we have $f_n \to f$ $a.e.$, could we say $f_n \to f$ in measure?

If those are not true, which conditions are required?

  • 2
    Please edit your post to show what you tried so far. Also, there is no complex or functional analysis involved here. Please remove those tags. You may consider adding the measure theory tag, however. – 2017-02-22

1 Answers 1

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  1. yes
  2. yes
  3. no

If the space is finite or if the convergence is dominated, then 3 is valid.