Pattab

2022-07-01

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

$\underset{x\to {c}^{+}}{lim}f(x)=L$

iff

$\mathrm{\forall}\u03f5,\mathrm{\exists}\delta >0,J:=f((c,c+\delta )),\mu (J\cap {B}_{\u03f5}(L))=\mu (J)$

$\underset{x\to {c}^{+}}{lim}f(x)=L$

iff

$\mathrm{\forall}\u03f5,\mathrm{\exists}\delta >0,J:=f((c,c+\delta )),\mu (J\cap {B}_{\u03f5}(L))=\mu (J)$

Danika Rojas

Beginner2022-07-02Added 9 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, ${A}_{1},\cdots ,{A}_{n}$ be subsets that partition $\mathbb{R}$, and ${\alpha}_{1},\cdots ,{\alpha}_{n}$ real numbers. Consider $f:=\sum _{i=1}^{n}{\alpha}_{i}{\mathbf{\text{1}}}_{{A}_{i}}$.

Then f verifies your condition everywhere, since $\mu (\{f(x)\text{}|\text{}x\in \mathbb{R}\})=0$.

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?

Let n be an integer, ${A}_{1},\cdots ,{A}_{n}$ be subsets that partition $\mathbb{R}$, and ${\alpha}_{1},\cdots ,{\alpha}_{n}$ real numbers. Consider $f:=\sum _{i=1}^{n}{\alpha}_{i}{\mathbf{\text{1}}}_{{A}_{i}}$.

Then f verifies your condition everywhere, since $\mu (\{f(x)\text{}|\text{}x\in \mathbb{R}\})=0$.

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

Beginner2022-07-03Added 5 answers

Great point about measurability issues. Also, I like your example function a lot. That's very useful.

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