# Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

1,606 questions
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### Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
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### Nonlinear maps in Riesz Thorin theorem

The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear. What I was wondering about is whether this is because otherwise you do ...
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### Can a big set always look small?

For a set $C\subset \mathbb R^2$, define its visibility from a point $x$ as $vis_C(x)=\{\varphi\in \mathbb S^1\mid \exists t>0~~x+t*\varphi\in C\}$, where $\mathbb S^1$ denotes the unit circle. Say ...
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### Closedness of the set of probability measures anihilating a measurable function

Let $X$ be a compact metrizable topological space and let $f$ be a bounded, real valued, Borel function on $X$. Denoting by $P(X)$ the collection of all probability measures on $X$, consider the ...
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### Antisymmetry of the stochastic order

An ordered topological space is a topological space $X$ equipped with a partial order $\leq$ which is closed as a subset of $X\times X$. By antisymmetry of $\leq$, it follows that the diagonal of $X$ ...
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### Non-probabilist term for conditional expectation?

When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about ...
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you. Let $X$ be a topological ...