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 is a probability measure on respect to ( i have supposed is a measurable space) and if is a function from to (real numbers set ,sorry for my poor Latex too) , if the set is finite , if is measurable respect to and ,if is a natural , such that where are real numbers and denotes the indicator function of and ,analogously is a natural , such that where are real numbers ,how can i prove that ? . My idea was to account the standard representation of f setting where ,natural , is the cardinality of , let so there is the standard representation of ,and i tried to show is equal to one between and .To make this i observed : where are fixed in .Intuitively it results ma i can't prove it.I hope someone help me ,thank you so much.