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:
$\underset{x\to {c}^{+}}{lim}f\left(x\right)=L$
iff
$\mathrm{\forall }ϵ,\mathrm{\exists }\delta >0,J:=f\left(\left(c,c+\delta \right)\right),\mu \left(J\cap {B}_{ϵ}\left(L\right)\right)=\mu \left(J\right)$

Danika Rojas

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 .

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