## Doctoral Theses

### Determination of Probability Measures Through Group Actions.

2-28-1979

2-28-1980

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

#### Supervisor

Parthasarthy, K. R. (TSMU-Delhi; ISI)

#### Abstract (Summary of the Work)

One of the fundamental problems in Measure Theory is the following: given a measurable space (x, B,), to find subclasses D of B, such that whenever for two probability measures u and v on (X, B,), u(B) = v(3) for every B c D, then u(B) = v(B) for every Be B,. The first basic theorem of Measure Theory, viz., the Caratheodory Extension Theorem says that any sub-algebra D of B, which generates B, has the above mentioned property.Let (X, B, ) be a given measurable space. A subclass n of 3, is called a determining class for a class P of probability measures on (X, B,) if for u,v E P, whenever u(B) = v(B) for every B e D, then u(B) = v(B) for every Be B,. The problem of finding determining classes, other than the ones assured by the Caratheodory Extension Theorem, has been of interest. One of the earlier results in this direction is due to Cramer and World [41. They considered the case X = R' and showed that the class D = (S t e , s e R) is a determining class for the class of S E all probability measures on , where for t = (t, ,...,t) e R&; and s e R, ; s). In the case, when X is a %3D t,s metric space and B, is the o-algebra of Borel subsets of X, various i=1 determining classes (which utilize the topoloy of X) are known for M(X), the class of all probability measures on (X, B,). For examine one knows that the class O of all open subsets of X is a determining class for M(X). If X is a complete separable metric space, then the class K of all compact subsets of X is a determining class for M(X ), Another class of subsets in a metric space X which is of interest is the class S of all closed balls in X. It turns out that, in general S is not a determining class for M(X). Davis [5] has shown that one can construct a compact metric space and two distinct Borel probability measures on it which agree on all closed balls, The problem off finding metric spaces X in which the class S is a determining class for M(X) has been investigated by many authors, for example : Besicovitch [2], Anderson [1], Christensen [3] and Hoffmann - Jorgensen (12].

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843477

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