This is a problem from my real analysis class
The Dirac measure $\delta$ concentrated on $\left\lbrace 0\right\rbrace$ is a Borel measure on $\mathbb{R}$. Find all the increasing and right-continuous functions $F: \mathbb{R} \rightarrow \mathbb{R}$ such that $\mu_F = \delta$
So from class, I know that $\mu_F\left((a,b] \right) = F(b) - F(a)$. I therefore think I need a function $F$ so that $F = 0$ if $0 \notin (a,b]$ and $F = 1$ if $0 \in (a,b]$. Thus it seems like a natural choice to take the function:
$$ F = \begin{cases} 0 & x \leq 0 \\ 1 & x > 0 \\ \end{cases} $$
I'm not feeling confident that this is the right function or if it's the only one (I'm having a difficult time going back in forth in my head between Borel measures and associated functions). I don't think it's the only one but I think we have a theorem stating any increasing, right continuous function $F$ has a unique Borel measure on $\mathbb{R}$, and if $G$ is another function, $F-G$ is a constant. I have a feeling I'm supposed to use that to modify the $F$ I picked, but I'm unsure how.