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djalil.chafai.net | ||
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www.mardahl.dk
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almostsuremath.com
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| | | | | I will give a proof of the measurable section theorem, also known as measurable selection. Given a complete probability space $latex {(\Omega,\mathcal F,{\mathbb P})}&fg=000000$, we denote the projection from $latex {\Omega\times{\mathbb R}}&fg=000000$ by $latex \displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\pi_\Omega\colon \Omega\times{\mathbb R}\rightarrow\Omega,\smallskip\\ &\displaystyle\pi_\Omega(\omega,t)=\omega. \end{array} &fg=000000$ By definition, if $latex {S\subseteq\Omega\times{\mathbb R}}&fg=000000$ then, for every $latex {\omega\in\pi_\Omega(S)}&fg=000000$, there... | |
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almostsuremath.com
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| | | | | The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities. In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space $latex {(\Omega,\mathcal F,{\mathbb P})}&fg=000000$. This consists... | |
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makezine.jp
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