Mathematics Prelims

October 27, 2008

The Lebesgue-Stieltjes Measure

Filed under: Analysis,Measure Theory — cjohnson @ 5:19 pm

Consider an increasing, right-continuous function F : \mathbb{R} \to \mathbb{R}.  We can measure the length of an interval I in \mathbb{R} with end points a and b (e.g., {}[a, b]) as

\displaystyle \ell_F(I) = F(b) - \lim_{x \to a^-} F(a).

(Capinski uses \ell_F(I) = F(b) - F(a) with the interval I restricted to being of the form (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 \mathbb{R}, call it \mu_F^*, as follows.

\displaystyle \mu_F^*(A) = \inf\left\{ \sum_{n=1}^\infty \ell_F(I_n) : A \subseteq \bigcup_{n=1}^\infty I_n \right\}

Where each I_n is a bounded interval.  Proceeding as we would in defining the usual Lebesgue measure on \mathbb{R}, we will let \mu_F be a measure on \mathcal{M}_F where

\displaystyle \mathcal{M}_F = \{ E \subseteq \mathbb{R} : \forall A \subseteq \mathbb{R}, \, \mu_F^*(A) = \mu_F^*(A \cap E) + \mu_F^*(A \cap E^\complement) \}

Now we’ve gone from an increasing, right-continuous function to a measure on \mathbb{R}.  Note that sets that were null with the Lebesgue measure, may not be anymore, depending on our choice of F.  For instance, if we have

\displaystyle F(X) = \left\{ \begin{array}{ll} x & : x < 0 \\ 1 + x &: x \geq 0 \end{array}\right.

Then \mu_F(\{0\}) = 1, though with the standard Lebesgue measure we have \mu(\{0\}) = 0.

It will be convenient to have the convention that if F is an increasing, right-continuous function that

\displaystyle \int_E g \, dF

is actually short-hand for the Lebesgue integral of g over E using the measure obtained from F as we’ve described above.  This is normally referred to as the Lebesgue-Stieltjes integral with integrator F.

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3 Comments »

  1. […] 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 | Reply

  2. 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 | Reply

    • 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 | Reply


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