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

gaiaecologicaq2 2022-07-02 Answered
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 > 0 , J := f ( ( c , c + δ ) ) , μ ( J B ϵ ( L ) ) = μ ( J )
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Answers (2)

gozaderaradiox5
Answered 2022-07-03 Author has 19 answers
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?

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2d3vljtq
Answered 2022-07-04 Author has 5 answers
Great point about measurability issues. Also, I like your example function a lot. That's very useful.

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