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The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail.

Take, for instance, Chung and Williams' textbook "Introduction to Stochastic Integration", 2nd edition (Birkhaeuser 2014). Section 1.3, titled "Functions of Bounded Variation and Stieltjes Integrals", lists facts, but does not prove them nor gives reference for further reading.

Where can I turn to (a textbook, online class notes, etc.) to fill in the gaps and read an orderly exposition of the subject matter that accounts for all the results pertinent to stochastic integration, and, in particular, those cited in Chung and Williams' section 1.3?

P.S. The reference need not be in English.

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    $\begingroup$ I find this may be useful: math.stonybrook.edu/~daryl/ls.pdf $\endgroup$ – Gordon May 19 '16 at 19:22
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    $\begingroup$ See also this book amazon.ca/…. $\endgroup$ – Gordon May 19 '16 at 20:16
  • $\begingroup$ @Gordon: Thanks. I'm familiar with the book and I've perused the pdf file you linked to. They're both a great start, but neither covers all the requisite facts. To give just one example (there are many more), consider Chung and Williams' statement: "If $g$ is right continuous and locally of bounded variation on all of $[0, \infty)$, then the measures $\mu$ for different intervals $[0, t]$ are consistent with one another, but they do not in general extend to a measure on $[0, \infty)$, unless $g$ is of bounded variation on $[0, \infty)$." This statement is not addressed in the file or the book. $\endgroup$ – Evan Aad May 19 '16 at 20:23
  • $\begingroup$ Consistency means that the restriction of the measure defined on a larger interval to a smaller interval is the same as the measure defined on the smaller interval. If $g$ is of bounded variation on $[0, \infty)$, then a measure can be defined on $[0, \infty)$. A general textbook may not discussed such fine properties, but it is worthy to work them out yourself. You can also ask them as individual questions. However, it appears to me these properties are self-evident. $\endgroup$ – Gordon May 20 '16 at 18:43
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Royden's "Real Analysis" is a standard textbook for a first year grad course in Real Analysis. It covers all the integration topics nicely. A more stochastically oriented book would be "Probability with Martingales" by Williams, which covers integration as well.

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