Lebesgue-Stieltjes measure construction on $\mathbb{R}^n$.

borel-measureslebesgue-measuremeasure-theoryreference-request

In Athreya's "Measure Theory and Probability Theory", it introduces Lebesgue-Stieltjes measure construction on $\mathbb{R}^n$ in general in the following way:

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ satisfy the following properties:

  1. Let $(x_1,x_2)\leq (y_1,y_2)$ iff $x_1\leq y_1$ and $x_2\leq y_2$.
    If $x\leq y$ in this partial order, then $f(y_1,y_2)-f(x_1,y_2)-f(y_1,x_2)+f(x_1,x_2)\geq 0$.
  2. $f$ is right-continuous for each coordinate.
  3. $x\leq y\ (\forall x_n\leq y_n)\implies f(x)\leq f(y)$ (this condition is omitted but I think this might be needed).

Then $\mu((x_1,x_2]\times (y_1,y_2])=f(y_1,y_2)-f(x_1,y_2)-f(y_1,x_2)+f(x_1,x_2)$ defines a locally finite Borel measure on $\mathbb{R}^2$. Conversely, any locally finite Borel measure is generated by such a function.

I have some questions about this construction:

  1. I want to see a complete proof of this construction in $\mathbb{R}^n$; the first condition can be defined analogously in $\mathbb{R}^n$. Even in the book, there was no complete proof of this general construction. I am also not sure whether the third condition can be omitted or not.
  2. I wonder if there is a one-to-one correspondence between such functions and locally finite Borel measures on $\mathbb{R}^n$.
    In one-dimensional case, a locally finite Borel measure corresponds to a right-continuous increasing function up to a constant. It seems like such a correspondence result, if it exists, should be more constrictive; $f(x,y)=x+y$ yields the zero measure, but any $f(x,y)=F(x)+G(y)$ does so as well.

Any reference or complete construction would be much appreciated.

Best Answer

$\newcommand\R{\mathbb R}$Concerning your question 1: A complete proof of this result (based on a discrete approximation) can be found, for instance, in Kallenberg's book "Foundations of Modern Probability", where it is given as Corollary 3.26. Another kind of arguments, using e.g. semirings, can be found e.g. in Example 1.54 in Probability Theory, A Comprehensive Course, by Klenke, where the Lebesgue measure over $\R^d$ is constructed; the construction of the Lebesgue–Stieltjes measure is quite similar.

Concerning your question 2: Clearly, if $\mu_f$ denotes the Lebesgue–Stieltjes measure for a right-continuous function $f\colon\R^d\to\R$ with nonnegative increments (in the natural sense, as defined e.g. in Kallenberg's book), and if $g\colon\R^d\to\R$ is such that \begin{equation} \label{star} \tag{$*$} g(x_1,\dotsc,x_d)=f(x_1,\dotsc,x_d)+\sum_{j=1}^d h_j(x_1,\dotsc,x_{j-1},x_{j+1},\dotsc,x_d) \end{equation} for some functions $h_j\colon\R^{d-1}\to\R$ and all $(x_1,\dotsc,x_d)\in\R^d$, then $\mu_g=\mu_f$. Vice versa, if $\mu_g=\mu_f$, then the (appropriate $d$-fold) increments of $g-f$ are identically zero, which implies \eqref{star}.

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