Consider an increasing, right-continuous function 
. We can measure the length of an interval 
in 
with end points 
and 
(e.g., ![{}[a, b]](http://l.wordpress.com/latex.php?latex=%7B%7D%5Ba%2C+b%5D&bg=ffffff&fg=000000&s=0 "{}[a, b]")
) as
.
(Capinski uses 
with the interval restricted to being of the form ![(a, b]](http://l.wordpress.com/latex.php?latex=%28a%2C+b%5D&bg=ffffff&fg=000000&s=0 "(a, b]")
, but I believe this gives the same measure in the end.)
Using this definition of the length of an interval, we can then construct an outer measure on , call it 
, as follows.
Where each 
is a bounded interval. Proceeding as we would in defining the usual Lebesgue measure on , we will let 
be a measure on 
where
Now we’ve gone from an increasing, right-continuous function to a measure on . Note that sets that were null with the Lebesgue measure, may not be anymore, depending on our choice of 
. For instance, if we have
Then 
, though with the standard Lebesgue measure we have 
.
It will be convenient to have the convention that if is an increasing, right-continuous function that
is actually short-hand for the Lebesgue integral of 
over 
using the measure obtained from as we’ve described above. This is normally referred to as the Lebesgue-Stieltjes integral with integrator .
[...] that is an increasing, right-continuous function with and . Using the ideas from the Lebesgue-Stieltjes measure article, we can have that gives us a measure and sigma-algebra on . Let and be the measure and [...]
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