In a text I was reading, I saw the author refer to a function f being μ-measurable. Does this mean something different than saying that f is measurable with respect to the σ-algebra that μ acts on, or does it imply something more about f?

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In a text I was reading, I saw the author refer to a function   f being   μ-measurable. Does this mean something different than saying that   f is measurable with respect to the   σ-algebra that   μ acts on, or does it imply something more about   f?

Mihevcekd · Accepted Answer

A general notion of   μ-measurability is as follow: suppose that   f  :  E  →  F where   (  E  ,      E    ,  μ  ) is a measure space and   F is a Banach space equipped with it Borel   σ-algebra       F  . Then we says that   f is   μ-measurable if and only if there is a sequence of simple functions   {      f    n        }          n      ∈              N             such that       f    n    →  f   μ-almost everywhere, where a function   g is simple if and only if1.   g  (  E  ) is a finite set2.       g          −      1        (  y  )  ∈      E   for each   y  ∈  F3. and   μ  (      g          −      1        (  F  ∖  {  0  }  )  &amp;lt;  ∞It can be shown that the above is equivalent to say that   f is Borel measurable (that is,       f          −      1        (  A  )  ∈      E   for every   A  ∈      F  ) and that there is some   μ-null set   N such that   f  (      N    ∁    ) is separable.Then, in particular,   f is   μ-measurable when it is complex valued, or take values in any Euclidean space.The above definition of   μ-measurability can be extended, in an obvious way, to the case where   f take values in the extended real line or in the Riemann sphere (or more generally to the one point compactification of any Euclidean space).

In a text I was reading, I saw the author refer to a function f being μ

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