The Stieltjes integral¶
The Lebesgue-Stieltjes integral is a type of integral specially useful in probability theory, because of the resemblance between the Stieltjes measures and probability measures. To introduce the concept consider a real valued random variable that takes values on a closed interval \(\left[a,b\right]\), this is for example the result of coin toss when catalogued as \(0\) or \(1\), the underlying probability space is formed by \(S=\left\lbrace H,T\right\rbrace\), \(\mathcal{A}=\left\lbrace \emptyset,S,\left\lbrace H\right\rbrace ,\left\lbrace T\right\rbrace \right\rbrace\) and a probability measure on \(\mathcal{A}\), a function \(\mu:\mathcal{A}\to\left[0,1\right]\) such that \(\mu\left(\left\lbrace H\right\rbrace \right),\mu\left(\left\lbrace T\right\rbrace \right)\geq0\), \(\mu\left(S\right)=\mu\left(\left\lbrace H\right\rbrace \right)+\mu\left(\left\lbrace T\right\rbrace \right)=1\) and \(\mu\left(\emptyset\right)=0\). The random variable is then a function \(f:S\to\mathbb{R}\) such that \(f\left(H\right)=0\) and \(f\left(T\right)=1\). It seems natural to ask what is the probability that \(f\left(s\right)=1\), it is of course given by \(\mu\left(T\right)\), in the same way can ask for the probability that \(f\left(s\right)\leq c\) for any value \(c\), the function that answers that question is called the cumulative distribution function. In this example we have:
Because the measure \(\mu\) is non-negative it is clear that \(F\) has to be a non-decreasing function, it is also continuous from the left, moreover it is possible to recover \(\mu\) from knowledge of \(F\):
The Stieltjes measure is a general way of looking at this last step. It treats the problem of inducing a measure from a non-decreasing right continuous function. The application to probability theory is apparent because we deal with the CDF of a random variable, and not directly with its probability measure, as we saw before it is this latter object the one that defines the expected value.
Now we turn to define formally the Stieltjes integral. Let \(F:\left[a,b\right]\to\mathbb{R}\) be a non-decreasing and right-continuous function. Let \(\mathcal{A}\) be an algebra of all subintervals of \(\left[\alpha,\beta\right)\) (including open, closed and half-open intervals). Define a measure on \(\mathcal{A}\) by:
where \(F\left(x+0\right)=\lim_{\epsilon\to0;\epsilon>0}F\left(x+\epsilon\right)\) is the right limit of \(F\) at \(x\).
Now consider the Lebesgue extension of \(m\), call it \(\mu_{F}\) and the \(\sigma\)-algebra of all \(\mu_{F}\)-measurable, call it \(\mathcal{A}_{F}\). This set contains all subintervals of \(\left[\alpha,\beta\right)\) and hence all the Borel sets of \(\left[\alpha,\beta\right)\).
Definition 25.1: Stieltjes measure
The measure \(\mu_{F}\) described above is called the (Lebesgue-)Stieltjes measure and \(F\) its generating function.
This concept is easily extended to the whole real line. Some examples show the generality of this type of measure:
Example 25.1
Let \(F\left(x\right)=x\), then the Stieltjes measure is nothing but the Lebesgue measure on the real line, that is, the extension of the concept of length of an interval.
Example 25.2
Let \(F\) be a jump function with discontinuity points \(\left\lbrace x_{1},x_{2},\ldots\right\rbrace\) and corresponding jumps \(\left\lbrace h_{1},h_{2},\ldots\right\rbrace\). The measure is of course:
Then every subset of \(\left[\alpha,\beta\right)\) is \(\mu_{F}\)-measurable because their measure depends only on countable points. Any set \(A\) has measure given by:
This number exists by assumption. A Stieltjes measure generated by a jump function is called a discrete measure. All discrete random variables have CDF that are jump functions.
Example 25.3
Let \(F\) be an absolutely continuous non-decreasing function on \(\left[\alpha,\beta\right)\). Absolutely continuous functions have a finite derivative a.e. let this derivative be \(f=F^{\prime}\). Then the Stieltjes measure \(\mu_{F}\) is defined for all Lebesgue measurable sets and:
clearly in this case \(\mu_{F}\left(\left\lbrace x\right\rbrace \right)=0\) because \(\left\lbrace x\right\rbrace\) has Lebesgue measure \(0\).
The result follows from Lebesgue theorem:
Theorem 25.1: Lebesgue
If \(F\) is absolutely continuous on \(\left[a,b\right]\) then the derivative \(F^{\prime}\) is integrable on \(\left[a,b\right]\) and:
Applying this theorem here we get:
Because \(f\) is non-negative and integrable wrt all Lebesgue-measurable subsets of \(\left[a,b\right]\left(\mathcal{B}_{\left[a,b\right]}\right)\) we know by proposition (Proposition 24.3) that
is a measure on \(\left(\left[a,b\right],\mathcal{B}_{\left[a,b\right]}\right)\) that coincides with \(m\), because the extension is unique we get that \(\mu_{F}\) is the Stieltjes measure we are looking for.
This type of measure is called absolutely continuous and is related to continuous random variables.
Now we can define the integral with respect to a Stieltjes measure:
Definition 25.2: Lebesgue-Stieltjes integral
Let \(\mu_{F}\) be Stieltjes measure with generating function \(F\), and let \(g\) be a \(\mu_{F}\)-measurable function, then the integral is defined as:
If \(\mu_{F}\) is discrete with \(F\left(x\right)=\underset{x_{n}\leq x}{\sum}h\left(x_{n}\right)\), then we have:
If \(\mu_{F}\) is absolutely continuous then:
As hinted above in probability Stieltjes measures arise naturally. Let \(\xi\) be a random variable and define \(F\left(x\right)=\Pr\left(\xi<x\right)\), then as noted above \(F\) is non-decreasing and continuous from the left, moreover \(F\left(-\infty\right)=0\) and \(F\left(\infty\right)=1\). The Lebesgue-Stieltjes measure allows us to define the expected value and variance of the random variable as:
These definitions are valid for discrete and continuous random variables.