Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I'd like to have a correct general understanding of the importance of measure theory in probability theory.
For now, it seems like mathematicians work with the notion of probability measure and prove theorems, because it automacially makes the theorem true, no matter if we work with discrete and continuous probability distribution. One could say that the Law of large numbers justifies the definition of expected value, by the way.
Basic Measure Theory Definitions
Is it right to say that probability using the general notion of probability measure saves work of mathematicians? What are the other advantages? Please correct me if I'm wrong, but I hope you get the idea of what sort of information I expect - it's the importance and role of measure theory in probability and answer to the question: are there theorems in probability that do not hold for general probability measure, but are true only for either discrete or continuous probability distribution?
If we can prove that no such theorems can exist, we can simply forget about the distinction between discrete and continuous distributions. If someone could come up with a clear, concise summary, I'd be grateful.
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I'm not an expert, so please take that into account. Since measure-theoretic axiomatization of probability was formulated by Kolmogorov, I think you'd be very much interested in this article. I had similar questions to you, and most of them were clarified after the reading - although I've also read Kolmogorov's original work after that. Then the idea was: it works well, what if we try to use this method much more often, and even say that this is the way to go actually?
When the work of Kolmogorov was first out, not every mathematician was agree with his claim to say the least. But you are somewhat right in saying that measure theory allows dealing with probability easier. It's like solving basic geometric problems using vector algebra. However, there are some things that hold for "continuous" measures only.
Of course, that does not hold for measures with atoms. Not a very probabilistic fact though. This super professional statement, overkills every old fashion probability theory about discrete , continuous or mixed distribution functions! Theorem 1. Sign up to join this community.
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The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Probability and measure theory Ask Question. Asked 3 years, 10 months ago. A measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
Advanced measure theory (BM)
A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set.
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A measure is called complete if every negligible set is measurable. Measures are required to be countably additive. However, the condition can be strengthened as follows. If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable ; examples of such sets include the Vitali set , and the non-measurable sets postulated by the Hausdorff paradox and the Banach—Tarski paradox.
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the signed real numbers is called a signed measure , while such a function with values in the complex numbers is called a complex measure.
Measures that take values in Banach spaces have been studied extensively. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination , while signed measures are the linear closure of positive measures.
Another generalization is the finitely additive measure , also known as a content. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory ; this is the theory of Banach measures. A charge is a generalization in both directions: it is a finitely additive, signed measure.
From Wikipedia, the free encyclopedia. Not to be confused with Metric mathematics. For the coalgebraic concept, see measuring coalgebra. Main article: Sigma-finite measure. Main article: s-finite measure.
Main article: Complete measure. Main article: Non-measurable set. Robert G. Bauer, H. Dudley, Real Analysis and Probability. Cambridge University Press. Folland, Gerald B. Fremlin, Measure Theory. Torres Fremlin. Duncan Luce and Louis Narens Munroe, Introduction to Measure and Integration.
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Addison Wesley. Bhaskara Rao and M. Silverman, trans. Dover Publications. Emphasizes the Daniell integral.
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