- #1

psie

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- TL;DR Summary
- In measure theory, if ##\mu## is a finite Borel measure on the real line, then ##F(x)=\mu((-\infty,x])## defines an increasing and right continuous function from ##\mathbb R\to\mathbb R##. I realize I don't have any firm definition at hand to check that it is right continuous (nor have I found any definition online). Hence let me present one.

Here's my definition I've been working on.

Comments? Suggestions for improvements?

Let ##f:A\subset\mathbb R\to\mathbb R## be given. If ##c\in\mathbb R## is a limit point of ##A^+=\{x\in A:x\ge c\}##, then ##f## is right continuous at ##c## iff $$\lim_{n\to\infty}f|_{A^+}(x_n)=f|_{A^+}(c),$$ for every sequence ##(x_n)## in ##A^+## such that ##x_n\to c## as ##n\to\infty## and where ##f|_{A^+}## is the restriction of ##f## to ##A^+##.

Comments? Suggestions for improvements?

**EDIT**: The reason I'm looking for a sequential characterization of right continuous is because the way you check that ##F## is right continuous is through $$F(x)=\mu((-\infty,x])=\mu\left(\bigcap_{n=1}^{\infty}(-\infty,x_n]\right)=\lim_{n\to\infty}\mu((-\infty,x_n])=\lim_{n\to\infty}F(x_n),$$for any sequence ##(x_n)## such that ##x_n\searrow x## as ##n\to\infty##.