Let P and Q be two probability measures defined on the measurable space ((, A), and suppose that P ( Q (i e , P Q and Q P) Let f dP d and g dQ d for some ( finite measure dominating both P and Q (e g , P Q), and let Z log (g f), where as always, log stands for the natural logarithm Then (with reference to Exercise 3), show that P Q ( 2(1 e ) 2P ( Z ) for every 0

Question: Let P and Q be two probability measures defined on the measurable space ((, A), and suppose that P ( Q (i.e., P < <

Let P and Q be two probability measures defined on the measurable space ((, A), and suppose that P ( Q (i.e., P << Q and Q << P). Let f = dP/dµ and g = dQ/dµ for some (-finite measure µ dominating both P and Q (e.g., µ = P + Q), and let Z = log (g/f), where as always, log stands for the natural logarithm. Then (with reference to Exercise 3), show that
||P - Q|| ( 2(1 - e - ε) + 2P (|Z| > ε) for every ε > 0.

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