In real functions, do we have a notion of one-sided measure theoretic limits? I want to define them with the following: lim x → c + f ( x ) = L iff ∀ ϵ , ∃ δ &gt; 0 , J := f ( ( c , c + δ ) ) , μ ( J ∩ B ϵ ( L ) ) = μ ( J )

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In real functions, do we have a notion of one-sided measure theoretic limits? I want to define them with the following:            lim              x        →                  c          +                      f    (    x    )    =    L  iff  ∀  ϵ  ,  ∃  δ  &amp;gt;  0  ,  J  :=  f  (  (  c  ,  c  +  δ  )  )  ,  μ  (  J  ∩      B    ϵ    (  L  )  )  =  μ  (  J  )

Danika Rojas · Accepted Answer

First of all, it should be noted that the image of a Borel set by a continuous function may not be Borel, so there might be measurability issues.Let n be an integer,       A    1    ,  ⋯  ,      A    n   be subsets that partition       R  , and       α    1    ,  ⋯  ,      α    n   real numbers. Consider   f  :=      ∑          i      =      1        n        α    i              1                      A        i            .Then f verifies your condition everywhere, since   μ  (  {  f  (  x  )     |     x  ∈      R    }  )  =  0.In fact, if you want to define something, I think you should first make a list of whatever your definition is supposed to imply. For example, should your definition of measure-continuity-on-the-left be verified for continuous functions in the usual sense?

Jameson Lucero · Accepted Answer

Great point about measurability issues. Also, I like your example function a lot. That&#039;s very useful.

In real functions, do we have a notion of one-sided measure theoretic limits? I want to define them

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