Define a function $$F(x) = \begin{cases} 1 \text{ if } x > 0\\ 0 \text{ if } x \leq 0 \end{cases}$$ This left continuous function induce a measure $\mu$ on $\mathbb{R}$, defined by $$ \mu(A) = \inf \Big\{ \sum_{i=1}^n F(a_i) - F(b_i) \mid A \subset \bigcup_{i=1}^{n}(a_i, b_i] \Big\} $$
what is the $\mu(A)$ for $A \subset \mathbb{R}$ when $A$ is measurable?
For an interval of the $(a, b]$, it is obvious that $\mu((a,b]) = 1$ if $b > 0$, and $\mu((a,b]) = 0$ if $b \leq 0$. But what about some more complicated cases?
My naive guess is that $\mu(A) = 0$ if $A$ contains a elements bigger than $0$, but I am not sure if it is correct.
$\mu$ is simply the 'delta measure at $0$': $\mu(A)=1$ if $0 \in A$ and $\mu(A)=0$ otherwise.