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Math Help - Simple Random Variable

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    Simple Random Variable

    Hi everyone,sorry for my poor english (i'm italian),my name is Stefano ; For a week i 'm trying to solve the following probability question:
    If P is a probability measure on A ' respect to A ( i have supposed  ( A A ') is a measurable space) and if f is a function from A to R (real numbers set ,sorry for my poor Latex too) , if the set \{ f (b) \mid b  \in A\} is finite , if f is measurable respect to  ( A A ') and  ( R Borel R ) ,if n is a natural \neq 0 , \forall i \in \{ 1,...,n\} B_{i}\in A'  such that \forall b\in A f(b)= \sum_ {i=1}^{n} d_{i}*I(B_ {i}(b)) where d_{i} are real numbers and I(B_ {i}) denotes the indicator function of B_{i} and ,analogously n' is a natural \neq 0 , \forall i \in \{ 1,...,n'\} C_{i}\in A'  such that \forall b\in A f(b)=  \sum_ {i=1}^{n'} e_{i}*I(C_ {i}(b)) where e_{i} are real numbers ,how can i prove that \sum_{i=1}^{n} d_{i}*P(B_{i}) =  \sum_{i=1}^{n'} e_{i}*P(C_{i}) ? . My idea was to account the standard representation of f setting \{ f (b) \mid b  \in A\} =\{k_{1},...,k_{r}\} where r  ,natural \neq 0 , is the cardinality of \{ f (b) \mid b  \in A\} , \forall i\in\{1,..,r \} let F_{i} =\{b \in A \mid f(b)=k_ {i}\}  so there is the standard representation of f f(b)=  \sum_ {i=1}^{r} k_{i}*I(F_ {i}(b)) ,and i tried to show   \sum_ {i=1}^{r} k_{i}*P(F_ {i}) is equal to one between   \sum_ {i=1}^{n} d_{i}*P(B_ {i}) and   \sum_ {i=1}^{n'} e_{i}*P(C_ {i}) .To make this i observed : k{1}*P(F_{1}) +...+k_{r}*P(F_{r}) = (d_{1}*IB_{1}(p_{1}) +...+d{n}*IB_{n}(p_{1})) * P(F_{1})  +...+(d_{1}*IB_{1}(p_{r}) +...+d{n}*IB_{n}(p_{r}) * P(F_{r})= d_{1}*[(P(F_{1})*IB_{1}(p_{1})+...+(P(F_{r}*IB_{1}(p_{r})] +...+d_{n}*[(P(F_{1}*IB_{n}(p_{1})+...+(P(F_{r}*IB_{n}(p_{r})] where p_{i} are fixed in F_{i} .Intuitively it results [(P(F_{1)}*IB_{i}(p_{1})+...+(P(F_{r})*IB_{i}(p_{r}  )] = P(B_{i}) ma i can't prove it.I hope someone help me ,thank you so much.
    Last edited by TheDifferentialProability; November 29th 2012 at 05:40 AM.
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