Constructing a Borel measure on an interval from left continuous bounded variation without Caratheodory's Theorem

Question

There is a problem from Rudin's Real Complex Analysis such that $$ f\in BV[a,b]\ \text {and left continuous} $$ $$\implies \exists \mu:\text {a Borel measure on [a,b] such that} \ \mu{[a,x)}=f(x)-f(a)$$ for $x\in [a,b]$. I know I can construct such a measure using the Caratheodory Theorem. The problem is, unfortunately (or fortunately?), Rudin does not introduce the Caratheodory Theorem. So I believe that the construction could be done without relying to the theorem. Is it so? If so, can you give the proof? Thanks and regards.