Let [imath]\pi[/imath] be a probability measure on a measure space [imath]S[/imath].
(a) Given a subset [imath]A \subset S[/imath] with [imath]\pi(A)>0[/imath], define the restriction probability measure [imath]\rho_{A}[/imath] determined on the measure space [imath]S[/imath] by the set [imath]A[/imath]. Hint: Be sure to give a formula for [imath]\rho_{A}(T)[/imath] for a measurable set [imath]T \subset S[/imath].
(b) Use your definition from part (a) above to prove the following [imath]{ }^{1}[/imath] law of partition: If [imath]A_{1}, A_{2}, \ldots, A_{k}[/imath] are (pairwise) disjoint measurable sets in [imath]S[/imath] with
[imath]\qquad \bigcup_{j=1}^{k} A_{j}=S \quad \text { and } \quad \pi\left(A_{j}\right)>0 \quad \text { for } j=1,2, \ldots, k[/imath]
and [imath]A[/imath] is any measurable subset of [imath]S[/imath], then
[imath]\qquad \pi(A)=\sum_{j=1}^{k} \rho_{A_{j}}(A) \pi\left(A_{j}\right)[/imath]
(a) Given a subset [imath]A \subset S[/imath] with [imath]\pi(A)>0[/imath], define the restriction probability measure [imath]\rho_{A}[/imath] determined on the measure space [imath]S[/imath] by the set [imath]A[/imath]. Hint: Be sure to give a formula for [imath]\rho_{A}(T)[/imath] for a measurable set [imath]T \subset S[/imath].
(b) Use your definition from part (a) above to prove the following [imath]{ }^{1}[/imath] law of partition: If [imath]A_{1}, A_{2}, \ldots, A_{k}[/imath] are (pairwise) disjoint measurable sets in [imath]S[/imath] with
[imath]\qquad \bigcup_{j=1}^{k} A_{j}=S \quad \text { and } \quad \pi\left(A_{j}\right)>0 \quad \text { for } j=1,2, \ldots, k[/imath]
and [imath]A[/imath] is any measurable subset of [imath]S[/imath], then
[imath]\qquad \pi(A)=\sum_{j=1}^{k} \rho_{A_{j}}(A) \pi\left(A_{j}\right)[/imath]
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