An algebra is a collection of subsets closed under finite unions and intersections. A sigma algebra is a collection closed under countable unions and intersections. Whats the difference between …

Dec 25, 2014 · Borel σ -algebras turn op if we are working on topological spaces. If X denotes a topological space and τ is its topology then the smallest σ -algebra that contains τ (in other …

Oct 21, 2021 · I know that a $\sigma$ -algebra is a suitable generalization of the notion of sample space, in the following sense: Consider a sample space $\Omega$ and a collection $\mathscr …

Aug 22, 2019 · The sigma algebra M(ψ) generated by a set ψ is the intersection of all sigma-algebras that contain ψ. As an example if X = {1, 2, …, 6}, ψ = {{2, 4}, {6}} then M(ψ) would be …

Apr 7, 2021 · The Borel $\sigma$ -algebra on $\mathbb {R^d}$, denoted by $\mathcal {B} (\mathbb {R^d})$ is the $\sigma$ -algebra generated by the collection of open sets of …

Jul 23, 2015 · The following is the definition of the product $\sigma$-algebra given in Gerald Folland's Real Analysis: Modern Techniques and Their Applications (pg. 22) (note that …

Aug 28, 2014 · The Wikipedia page for $\\sigma$-algebra says this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on measure theory, where …

Dec 20, 2014 · Now remember, a $\sigma$-algebra is a collection of subsets of $\Omega$ satisfying some rules. The rule that (A.3) captures is that if you take countably many subsets …

Dec 28, 2025 · In simple terms, a $\\sigma$-algebra is the collection of all of the things we know how to measure. Why don't we call it something that more directly suggests this, for example a …

Oct 11, 2012 · To obtain the smallest $\sigma$-algebra containing it, all you need to do is add the missing sets that make it a $\sigma$-algebra (instead of just being a set).