Measure theoretically equivalent

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Two or more measures are said to be measure theoretically equivalent if they assign the same values to all measurable sets, meaning they agree on the outcomes of events. This concept often relates to mutual absolute continuity, but it encompasses broader conditions. The term is not widely recognized in literature, leading to confusion among those seeking clarity. The discussion highlights the difficulty in finding clear definitions and examples in existing texts. Understanding this concept is crucial for deeper insights into measure theory and its applications.
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When are two or more measures said to be measure theoretically equivalent?
I have spent sometime on searching it; I am not getting it. Please someone help.
 
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Well, start with the definition in your text, what is that?
(Since it is not widely-known terminology, we can only guess at it.)
 
I couldn't find this terminology in books. When I was reading a paper I-divergence geometry of distributions by I.CSISZAR, I came across this term. Initially I thought it could be mutually absolutely continuous; but its not. I couldn't make any guess.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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