Let alpha(t) be a non-decreasing function on B and consider the integral. int_(-oo)^(+oo) e^(-xt) d alpha(t) absolutely convergent in I.



Answered question


Let α ( t ) be a non-decreasing function on B and consider the integral
+ e x t d α ( t )
absolutely convergent in I.
Does exist a measure μ α related to the non-decreasing function α ( t ) and how it can be constructed?

Answer & Explanation



Beginner2022-09-08Added 7 answers

The unboundedness of the interval does not make much difference: we can still associate measures to nondecreasing functions. Indeed, the intervals [a,b) with < a b < form a semiring which generates the Borel σ-algebra on R . Given α, define the pre-measure μ α ( [ a , b ) ) = α ( b ) α ( a ). Then extend it by the Carathéodory's extension theorem to a Borel measure.
Alternatively, you can try a gluing argument. On each interval (−n,n) we have a measure μ n constructed from α. These measures are consistent in the sense that μ n ( A ) = μ m ( A ) if A ( n , n ) and n < m. It takes some work to verify that μ ( A ) = lim n μ n ( A ( n , n ) ) is a Borel measure on R

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