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almostsuremath.com | ||
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jmanton.wordpress.com
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| | | | | If $latex Y$ is a $latex \sigma(X)$-measurable random variable then there exists a Borel-measurable function $latex f \colon \mathbb{R} \rightarrow \mathbb{R}$ such that $latex Y = f(X)$. The standard proof of this fact leaves several questions unanswered. This note explains what goes wrong when attempting a "direct" proof. It also explains how the standard proof... | |
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terrytao.wordpress.com
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| | | | | Thus far, we have only focused on measure and integration theory in the context of Euclidean spaces $latex {{\bf R}^d}&fg=000000$. Now, we will work in a more abstract and general setting, in w... | |
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extremal010101.wordpress.com
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| | | | | Suppose we want to understand under what conditions on $latex B$ we have $latex \begin{aligned} \mathbb{E} B(f(X), g(Y))\leq B(\mathbb{E}f(X), \mathbb{E} g(Y)) \end{aligned}$holds for all test functions, say real valued $latex f,g$, where $latex X, Y$ are some random variables (not necessarily all possible random variables!). If $latex X=Y$, i.e., $latex X$ and $latex Y$ are... | |
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xorshammer.com
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| | | In the book A=B, the authors point out that while the identity $latex \displaystyle{\sin^2(|10 + \pi x|) + \cos^2(|10 + \pi x|) = 1}$ is provable (by a very simple proof!), it's not possible to prove the truth or falsity of all such identities. This is because Daniel Richardson proved the following: Let $latex \mathcal{R}$... | ||
