"Measure Theory and Probability Theory" by Athreya and Lahiri 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:

- 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 $f(y_1,y_2)-f(x_1,y_2)-f(y_1,x_2)+f(x_1,x_2)\geq 0$.
- $f$ is right-continuous for each coordinate.
- $x\leq y\ (\forall x_n\leq y_n)\implies f(x)\leq f(y)$ (this condition is omitted in the book, 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:

- 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.
- I wonder if there is a one-to-one correspondence between such functions and locally finite Borel measures on $\mathbb{R}^n$. In the 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 restrictive; $f(x,y)=x+y$ 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.

(The question was originally asked in stackexchange.)

[Edited: Added the authors.]