This could totally be a stupid question but I'm unsure: is a measure (ie positive, countable additive on a σ algebra, 0 for the empty set) actually a measurable function (wrt to the Borel-sigma algebra on R )?

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This could totally be a stupid question but I&#039;m unsure: is a measure (ie positive, countable additive on a   σ algebra, 0 for the empty set) actually a measurable function (wrt to the Borel-sigma algebra on       R  )?

esperoanow · Accepted Answer

For a measure space   (  X  ,      F    ,  μ  ) you have that      μ          −      1        (  (  −  ∞  ,  c  ]  )  =  {  A  ∈      F    :  μ  (  A  )  ⩽  c  }so you will need a   σ-algebra defined in       F   to define the measurability of   μ. Well, you can define this   σ-algebra using   μ, this will give an induced   σ-algebra in       F  , and we can note it by   σ  (  μ  ).

2d3vljtq · Accepted Answer

Given a function, a sigma algebra on the domain and a sigma algebra on the codomain, you can ask whether the function is measurable with respect to the given sigma algebras. You have only specified a sigma algebra on the codomain, so we can not answer your question.What probably confused you is the fact that the domain of a measure is a sigma algebra on some set, so we need to consider a sigma algebra on another sigma algebra - the domain of the measure - to determine whether it is measurable.

This could totally be a stupid question but I'm unsure: is a measure (ie positive, countable additiv

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