A Relativist's Toolkit - The Math of Black Hole Mechanics by E. Poisson

By E. Poisson

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Proof For a quick proof we can apply Caratheodory's extension theorem. We need to check <7-additivity cr-additivity of °M on A. Suppose that (A n ), n g IN, is a sequence of pairwise disjoint sets from A such that U An € A. 4) there is m eGl N IN such that m IJ U AAn- = | J An. neIN neIN n=l Ti = l So Ak = = 0 for k > > m, and A M A M A n) =E £°M(A„)=£°M(A °M VneIN ( U A «)/ = °M (U -) = ° ( ") = E ° ( «) \n=l / n=l nglN VneIN / \n=l / n=l ngIN using finite additivity. It is quite straightforward to give an alternative 'first principles' proof which is illuminating and provides extra information.

We need to check <7-additivity cr-additivity of °M on A. Suppose that (A n ), n g IN, is a sequence of pairwise disjoint sets from A such that U An € A. 4) there is m eGl N IN such that m IJ U AAn- = | J An. neIN neIN n=l Ti = l So Ak = = 0 for k > > m, and A M A M A n) =E £°M(A„)=£°M(A °M VneIN ( U A «)/ = °M (U -) = ° ( ") = E ° ( «) \n=l / n=l nglN VneIN / \n=l / n=l ngIN using finite additivity. It is quite straightforward to give an alternative 'first principles' proof which is illuminating and provides extra information.

The intersection is treated in the same way. © ^ is a sequence of internal sets (indexed by IN not *IN). Suppose now that (An)ne n^i (JnglN An is not internal, nor Then — perhaps contrary to intuition — in general (JngiN (\giN An- This is an extremely important property called Hi-saturation. There is HngiN are many equivalent formulations of this property. First we present it as a type of compactness property of internal sets. 3 (Hi-Saturation) Let (An)ne¥i be a sequence of internal subsets of an internal set A.

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