Let's say we're given a Lebesgue-Stieltjes measure &#x03BC;<!-- μ --> and an associated functi

Nickolas Taylor 2022-07-11 Answered
Let's say we're given a Lebesgue-Stieltjes measure μ and an associated function F μ such that
μ ( [ a , b ) ) = F μ ( b ) F μ ( a )
for all semi-intervals in R .
I aim to calculate μ ( [ a , b ] ) , μ ( ( a , b ) ) , μ ( ( a , b ] ) in terms of the function F μ .
My current idea is to express the closed, open, and semi-interval in terms of the left semi-interval as such:
( a , b ) = i = 1 [ a i , b )
where a < a i + 1 < a i and lim n a n = a. For example, such a sequence could be a i = a + 2 i .
Then, μ ( ( a , b ) ) = i = 1 μ ( [ a i , b ) ) = i = 1 ( F μ ( b ) F μ ( a + 2 i ) ) .
We could get similar expressions using ( a , b ] = [ a ϵ , b ] [ a ϵ , a ] and then substituting as needed.
However, all of these expressions would be extremely messy and would give very complicated answers. Is there a cleaner way of expressing the measure of each interval?
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Answers (1)

Nathen Austin
Answered 2022-07-12 Author has 14 answers
μ ( ( a , b ) ) = lim n μ ( [ a + 1 n , b ) ) = lim n [ F μ ( b ) F μ ( a + 1 n ) ]. This is F μ ( b ) F μ ( a + ) where F μ ( x + ) denotes the right-hand limit of F μ ( b ) at x.
Similarly, F μ ( [ a , b ] ) = F μ ( b + ) F μ ( a ) and F μ ( ( a , b ] ) = F μ ( b + ) F μ ( a + ).
[It should be noted that F μ is continuous from the left at every point and, as a monotone function, has right hand limit at every point].
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