Math 2.0

definition

Borel sigma-algebra

Definition

Let XX be a topological space. The Borel σ-algebra B(X)\mathcal{B}(X) is the smallest σ-algebra on XX that contains each open subset of XX. It is the intersection of all σ-algebras on XX that contain the open sets. A Borel set of XX is an element of B(X)\mathcal{B}(X).

Remarks

On the real line, B(R)\mathcal{B}(\mathbb{R}) is also the σ-algebra generated by the open intervals (a,b)(a, b). The half-lines (−∞,b](-\infty, b] with b∈Rb \in \mathbb{R} also generate it. Each open set, each closed set, each countable set and each interval of R\mathbb{R} is a Borel set. Some subsets of R\mathbb{R} are not Borel sets. For example, the Vitali sets are not Lebesgue measurable, so they are not Borel sets.

Explanations

L1 human

We want to give a probability to intervals such as "the waiting time is between 22 and 55 minutes". The Borel sets are all the sets that we can build from intervals with complements and countable unions, again and again. So they are the events that come from intervals.

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  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Borel_set, Generating the Borel algebra

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