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.

 

[tacman]

To show that f satisfies the given equation, we need to show that for any set S in the Borel sigma-algebra B(I), the measure of S under m is equal to the Lebesgue measure of f(S).

First, we note that m is a non-atomic measure on the Borel sigma-algebra B(I), which means that for any set S in B(I), m(S) can be written as the sum of the measures of its individual points. This is because non-atomic measures do not assign positive measure to single points.

Next, we consider the function f(x) = m(I intersect(-infinity, x]). This function gives us the measure of the interval I intersecting with the interval (-infinity, x].

Now, for any set S in B(I), we can write it as the union of disjoint intervals, say S = U_n I_n, where I_n are the individual intervals and U_n is the union.

Then, we have f(S) = m(I intersect(-infinity, U_n I_n]) = m( U_n(I intersect(-infinity, I_n])) = m(U_n) = m(S).

Therefore, we have shown that for any set S in B(I), the measure of S under m is equal to the Lebesgue measure of f(S). This proves that f satisfies the given equation.