In measure theory I encountered Egorov's theorem which states that if ( X , S , μ ) is a measure space such that μ ( X ) &lt; ∞ i.e. μ is a finite measure.If ( f n ) be a sequence of measurable functions on X converging pointwise to f : X → R ,then f n is almost uniformly convergent to f.Now in the book the definition of almost uniform convergence is the following: f n → f almost uniformly if for each ϵ &gt; 0,there exists E ϵ ⊂ X such that μ ( E ϵ ) &lt; ϵ and f n → f uniformly on X − E ϵ .Now in some books I have seen that almost everywhere means outside a measure 0 set.So the definition of almost uniformly convergent should have been f n → f almost uniformly if ∃ E ⊂ X such that μ ( E ) = 0 and f n → f uniformly on X − E.But unfortunately the definition is not so.In fact the latter condition is stronger.I want to know why the former is taken as a definition and what the problem with the latter one is.I want to understand where I am making mistake in understanding the word "almost".

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In measure theory I encountered Egorov&#039;s theorem which states that if   (  X  ,      S    ,  μ  ) is a measure space such that   μ  (  X  )  &amp;lt;  ∞ i.e.   μ is a finite measure.If   (      f    n    ) be a sequence of measurable functions on   X converging pointwise to   f  :  X  →      R  ,then       f    n   is almost uniformly convergent to   f.Now in the book the definition of almost uniform convergence is the following:      f    n    →  f almost uniformly if for each   ϵ  &amp;gt;  0,there exists       E    ϵ    ⊂  X such that   μ  (      E    ϵ    )  &amp;lt;  ϵ and       f    n    →  f uniformly on   X  −      E    ϵ  .Now in some books I have seen that almost everywhere means outside a measure 0 set.So the definition of almost uniformly convergent should have been       f    n    →  f almost uniformly if   ∃  E  ⊂  X such that   μ  (  E  )  =  0 and       f    n    →  f uniformly on   X  −  E.But unfortunately the definition is not so.In fact the latter condition is stronger.I want to know why the former is taken as a definition and what the problem with the latter one is.I want to understand where I am making mistake in understanding the word &quot;almost&quot;.

Sanaa Hinton · Accepted Answer

Your alternate definition wouldn&#039;t work.Here is a counterexample:       u    n    (  x  )  =      x    n   on [0,1], which converges pointwise to 0 if   x  &amp;lt;  1 and 1 otherwise. This sequence doesn&#039;t converge uniformly on [0,1], nor on the half-open set [0,1), and you can&#039;t find a set   S of measure 0 such that convergence is uniform on   [  0  ,  1  ]  ∖  S: you would still need values of   x arbitrarily close to 1.However,       u    n   converges uniformly on every compact interval that doesn&#039;t contain 1, so you can find       S    m    =  (  1  −  1      /    m  ,  1  ] such that   μ  (      S    m    )  →  0, and       u    n   converges uniformly on   [  0  ,  1  ]  ∖      S    m   for all   m.

In measure theory I encountered Egorov's theorem which states that if ( X , <mrow class

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