measurable space

A measurable space is a set $E$ together with a collection $\\mathcal{B}$ of subsets of $E$ which is a sigma algebra.

The elements of $\\mathcal{B}$ are called measurable sets.

A measurable space is the correct object on which to define a measure; $\\mathcal{B}$ will be the collection of sets which actually have a measure. We normally want to ensure that $\\mathcal{B}$ contains all the sets we will ever want to use. We usually cannot take $\\mathcal{B}$ 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 $\\mathcal{B}$ cannot be the collection of all subsets if we hope to have a measure that returns the length of an open interval.

单词	MeasurableSpace
释义	measurable space A measurable space is a set $E$ together with a collection $\\mathcal{B}$ of subsets of $E$ which is a sigma algebra. The elements of $\\mathcal{B}$ are called measurable sets. A measurable space is the correct object on which to define a measure; $\\mathcal{B}$ will be the collection of sets which actually have a measure. We normally want to ensure that $\\mathcal{B}$ contains all the sets we will ever want to use. We usually cannot take $\\mathcal{B}$ 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 $\\mathcal{B}$ cannot be the collection of all subsets if we hope to have a measure that returns the length of an open interval.
随便看	IncreasingdecreasingmonotoneFunction increasing/decreasing/monotone function IndecomposableGroup indecomposable group IndefiniteAndDefiniteSums indefinite and definite sums IndefiniteSum indefinite sum IndependenceOfCharacteristicPolynomialOnPrimitiveElement independence of characteristic polynomial on primitive element IndependenceOfPadicValuations independence of p-adic valuations IndependenceOfValuations independence of valuations Independent independent IndependentIdenticallyDistributed independent identically distributed IndependentIncrement independent increment IndependentSetAndIndependenceNumber independent set and independence number IndependentSigmaAlgebras independent sigma algebras IndependentStochasticProcesses