Tags » Sigma-algebra

Clarification on the countable additivity of Lebesgue measure

As emphasized in remark 1.2.4, p. 19, of Terence Tao’s book “An introduction to measure theory”, finite additivity doesn’t hold for Lebesgue outer measure in general, and therefore it doesn’t hold for Lebesgue measure either. 480 more words

Mathematics

Clarification on the proof of Carathéodory's lemma

Carathéodory’s lemma, as it appears in p. 197 (appendix A1) of the book “Probability with martingales” by David Williams, is stated as follows:

Let be an outer measure on the measurable space… 626 more words

Mathematics

Clarification on algebras and σ-algebras of a set

David Williams makes a terminological remark on algebras in p. 16 of his book “Probability with martingales”; he mentions that an algebra on a set (defined ordinarily as a collection of subsets of that contains and that is closed under set complementation and closed under finite set unions) is “a true algebra in the algebraists’ sense”. 168 more words

Mathematics

Stochastic Process

Definition

Stochastic Process

A stochastic process is defined to be a function that maps the set to random variable .

State Space

Let and be the values of the random variables and B be its…

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