34. Let P be a probability measure on a finite probability space . In this problem...

 

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34. Let P be a probability measure on a finite probability space . In this problem we allow the possibility that P(w) = 0 for some values of w N. Let Z be a random variable on 2 with the property that P(Z > 0) = 1 and EZ = 1. For w, define P(w)Z(w) = P(w) and for some event , define P(A) = P(w). == Show the following a. What is the definition of a probability measure. b. Show that P(2) = 1. (note that P is a probability measure). c. If A is an event with P(A) = 0, then P(A) = 0. d. Assume that P(Z > 0) = 1. Show that if A is an event with P(A) = 1 then P(A) = 1. 34. Let P be a probability measure on a finite probability space . In this problem we allow the possibility that P(w) = 0 for some values of w N. Let Z be a random variable on 2 with the property that P(Z > 0) = 1 and EZ = 1. For w, define P(w)Z(w) = P(w) and for some event , define P(A) = P(w). == Show the following a. What is the definition of a probability measure. b. Show that P(2) = 1. (note that P is a probability measure). c. If A is an event with P(A) = 0, then P(A) = 0. d. Assume that P(Z > 0) = 1. Show that if A is an event with P(A) = 1 then P(A) = 1.