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How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?

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    But there is always a metric such that convergence in measure iff convergence with respect to the metric.2011-12-01
  • 3
    See also [this MO thread](http://mathoverflow.net/questions/5537/notions-of-convergence-not-corresponding-to-topologies) and [this blog post](http://chromotopy.org/?p=354).2011-12-01

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