# measure associated with increasing and right continuous function

let $F$ be increasing and right continuous and let $\mu_{F}$ be the associated measure .

Then for example:

$\mu_{F}\{a\}=F(a)-F(a-)$

$\mu_{F}[a,b]=F(b)-F(a-)$

Solution.

we let $a<b$ and since F is increasing and $F(a)<\infty$ the limit $F(a)$ exists and $\displaystyle\lim_{n\to\infty}F(a-\frac{1}{n})=F(a-)$

My question is: what does $F(a-)$ mean? value of the function at.... ??? is it limit at $a$ from the left side?

why not

$\displaystyle\lim_{n\to\infty}F(a-\frac{1}{n})=F(a)$???

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"is it limit at $a$ from the left side?" Precisely that. Since $F$ need not be continuous, the limit can be different from $F(a)$. –  Daniel Fischer Oct 9 '13 at 12:44
@DanielFischer: thank you! –  J.W.S Oct 9 '13 at 13:04