How Does a Non-Atomic Measure Relate to Lebesgue Measure Through a Function?

[hj2000]

if m(.) is a non-atomic measure on the Borel sigma-algebra B(I).
I is some fixed closed finite interval in R.

How to show that f satisfies the following:

m(S) = L(f(S)), S in B(I) where L is the Lebesgue measure and
f(x) = m( I intersect(-infinity,x] )

 

[EnumaElish]

You might get a reply if you use the standard homework template.