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A {\em measurable space} is a set $E$ together with a collection $\mathcal{B}(E)$ of subsets of $E$ which is a sigma algebra. |
| A \emph{measurable space} is a set $E$ together with a collection $\mathcal{B}(E)$ of subsets of $E$ which is a sigma algebra. |
The elements of $\mathcal{B}(E)$ are called {\em measurable sets}. |
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| The elements of $\mathcal{B}(E)$ are called \emph{measurable sets}. |
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| A measurable space is the correct object on which to define a measure; $\mathcal{B}(E)$ will be the collection of sets which actually have a measure. We normally want to ensure that $\mathcal{B}(E)$ contains all the sets we will ever want to use. We usually cannot take $\mathcal{B}(E)$ to be the collection of all subsets of $E$ because the axiom of choice often allows one to construct sets that would lead to a contradiction if we gave them a measure (even zero). For the real numbers, Vitali's theorem states that $\mathcal{B}(E)$ cannot be the collection of all subsets if we hope to have a measure that returns the length of an open interval. |
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