existence of countably additive measure for Borel subsets of $[0,1]$

Problem: Let $F(x)$ be a continuous and non-decreasing function on $[0,1]$ with $F(0)=0$ and $F(1)=1$. I’d like to prove the existence of a couuntably additive measure $mu$ on Borel subsets of $[0,1]$ s.t. $muleft([a,b]right)=F(b)-F(a)$ for all intervals $[a,b]subset[0,1]$.

I can prove the case when $F(x)$ is the distribution function, which is non-decreasing and right continuous on the real line. Then, $F(x)=muleft((-infty,x]right)$. $muleft((a,b]right)=F(b)-F(a)$.

My sketch of the proof for this case: We need to show that if $(a,b]=bigcup_{j=1}^{infty}(a_j,b_j]$, then $F(b)-F(a)=sum_{j}F(b_j)-F(a_j)$.

$ge$ is obvious.

For $leq$, we apply Caratheodory extension for $mu$ from the semiring of intervals to the Borel $sigma$-field. By definition of distibution function, $F(x)to 0$ as $xto-infty$. We can replace $a$ by finite number of $a’$ s.t. $F(a’)-F(a)<varepsilon$. Similary, we can replace $b$ by finite $b’$ s.t. $F(b)-F(b’)<varepsilon$. By right continuity of $F(x)$, we can replace $(a_j,b_j]$ by $(a_j,b_j’)$ s.t. $F(b_j’)-F(b_j)<frac{varepsilon}{2^j}$. Now we have
$$F(b’)-F(a’)ge F(b)-F(a)-2varepsilon$$ and $[a’,b’]subset(a,b]$ is closed and bounded.

Moreover, $(a_j,b_j’)$ is an open cover of $[a’,b’]$. By Heine-Borel, we can find such finite subcover and
$$sum_{j}F(b_j)-F(a_j)gesum_{j}F(b_j’)-F(a_j)-frac{varepsilon}{2^j}ge F(b’)-F(a’)-2varepsilonge F(b)-F(a)-3varepsilon$$
Let $varepsilonto 0$ to get the desired result.

My questions are:

1. How can I adapt the proof I provided above for the problem I am asking? I am stuck, since countable union of disjoint closed sets may not be a closed set. And the problem seems to be a restriction of the distribution function to the interval $[0,1]$. If so, how? And if not, how can I write a new proof?

2. I am also interested in the case when $F(x)$ has jumps on its domain $[0,1]$. Do I have break into the case on finite and infinite jumps? I know the fact that Dirac measures are countably additive, so does this fact help to prove this case?

Added(09/13/2020): I think it is possible for singular points to have non-zero measures. For example in my case, we can define Dirac measure at $x=frac{1}{2}$, a valid countably additive measure on $mathcal{B}$. $$F(x)=chi_{{[1/2,1]}}$$ is one such example. Please point out if I have any flaw in my reasoning.

If there is a similar question regarding this, please direct me to that. I know this may be a basic question, but I’d like to receive help on this. Thank you.