Previously, a probability measure was applied to the event space which satisfies three properties and provides information about the sizes of those events. These properties are known as the three axioms of probability. The Vitali set is an example of a subset of that is not measurable. On the other hand, the Borel σ-field is measurable, which is easily demonstrated for the Lebesgue measure.

Example 3.1 (Borel set). Let [a, b] for be a closed interval in . Then (i) and (ii) . (iii) If [c, d] for is another interval such that , then .

Definition: Measurable Set A

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