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 &lt; a i + 1 &lt; 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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Let&#039;s say we&#039;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  &amp;lt;      a          i      +      1        &amp;lt;      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?

Nathen Austin · Accepted Answer

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].

Let's say we're given a Lebesgue-Stieltjes measure μ and an associated functi

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