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- 8/8/10
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Hi guys!
I would like to introduce my notes on measure theory and its interplay with stochastic processes: Yet another, yet very reader-friendly, introduction to the measure theory
I try to make subject as clear as possible, first I consider the simplest case: measure on (0,1) – as Lebesgue himself did. For this case I show how semirings, rings and sigma-algebra naturally arise.
Then we consider a measure on an abstract semiring and its extension to a sigma-algebra (Caratheodory construction). A reader will permanently notice similarities of a concrete case “(0,1) with Lebesgue measure” and abstract measure space. I also show why a semiring is important as a starting point and why it must be supplied with a sigma-additive measure.
Last but not least I informally apply Caratheodory construction to show how to get a sigma algebra generated by Wiener Process. This is my own pedagogical innovation, I read lots of books on measure theory and probability but have not met anything similar.
I also notice that the books on real analysis dwell on nuances like non-measurable sets, completeness of a measure and so on but lack for interplay with probability. And the books on probability tend to start with Caratheodory construction, which is very elegant but totally unmotivated, if presented as such. I tried to avoid both of these extremes.
I would like to introduce my notes on measure theory and its interplay with stochastic processes: Yet another, yet very reader-friendly, introduction to the measure theory
I try to make subject as clear as possible, first I consider the simplest case: measure on (0,1) – as Lebesgue himself did. For this case I show how semirings, rings and sigma-algebra naturally arise.
Then we consider a measure on an abstract semiring and its extension to a sigma-algebra (Caratheodory construction). A reader will permanently notice similarities of a concrete case “(0,1) with Lebesgue measure” and abstract measure space. I also show why a semiring is important as a starting point and why it must be supplied with a sigma-additive measure.
Last but not least I informally apply Caratheodory construction to show how to get a sigma algebra generated by Wiener Process. This is my own pedagogical innovation, I read lots of books on measure theory and probability but have not met anything similar.
I also notice that the books on real analysis dwell on nuances like non-measurable sets, completeness of a measure and so on but lack for interplay with probability. And the books on probability tend to start with Caratheodory construction, which is very elegant but totally unmotivated, if presented as such. I tried to avoid both of these extremes.