Let ( Ω , μ ) be a measure space. Suppose that ( f n ) converges almost everywhere to some function f where each f n belongs to L ∞ ( Ω ). Suppose that sup n ‖ f n ‖ ∞ &lt; ∞.Do we have f ∈ L ∞ ( Ω ) ?

Question

Let   (  Ω  ,  μ  ) be a measure space. Suppose that   (      f    n    ) converges almost everywhere to some function   f where each       f    n   belongs to       L    ∞    (  Ω  ). Suppose that       sup    n    ‖      f    n        ‖    ∞    &amp;lt;  ∞.Do we have   f  ∈      L    ∞    (  Ω  ) ?

Yair Boyle · Accepted Answer

Yes.Let   C  =  sup  ‖      f    n        ‖    ∞   and denote by       Ω    n   the set of   x for which       |        f    n    (  x  )      |    ≤  ‖      f    n        ‖    ∞    ≤  C. Let       Ω    ∞   be the set of   x for which       f    n    (  x  ) converges to   f  (  x  ).Then we have for   x  ∈      Ω    ∞        ⋂    n        Ω    n  ,      |    f  (  x  )      |    =      lim    n        |        f    n    (  x  )      |    ≤  Cand the set       Ω    ∞        ⋂    n        Ω    n   is a set with complement of zero measure.

Let ( <mi mathvariant="normal">Ω , μ ) be a meas

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