Consider an increasing, right-continuous function . We can measure the length of an interval in with end points and (e.g., ) as

.

(Capinski uses with the interval restricted to being of the form , 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 .

### Like this:

Like Loading...

*Related*

[…] 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 […]

Pingback by Distribution of a Random Variable « Mathematics Prelims — November 1, 2008 @ 12:27 pm |

The first line seems wrong.

The function is defined as right continuous. x tends to a+ not a- .

Comment by N. Srinivasan — April 13, 2010 @ 9:08 am |

F(a+)=F(a) by right continuity, and so, we do not need F(a+); but we need F(a-), since F(b)-F(a) is the measure of (a,b] rather than [a,b]; in fact, F(a)-F(a-) is the measure of {a}=[a,a].

Comment by Boris Tsirelson — February 29, 2012 @ 8:46 am |