I am looking for a book dealing with $n$-dimensional Lebesgue-Stieltjes Measure, especially dealing with its construction and its extension to Borel $\sigma$-algebra on $R^d$ rigorously.
For example, to show the extension to the Borel $\sigma$-algebra exists, we need to show that the set function, defined on the semialgebra consisting of all left-open and right-closed rectangles, is finitely additive and countably subadditive. Many books mention this result but left it as an exercise as they state that it is the same case as 1-dimensional. However, I think the case on $n$-dimensional is much more complex.
So I want a book that deals with such problems in a clear, complete and rigorous way, without referring to any exercise. The books can be either in classical real analysis or regarding probability theory or stochastic calculus. Thanks a lot in advanced!