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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Verifying that a map to $L^2_{\text{loc}}$ is continuous

Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
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Differentiation on $[0,1]$

EDIT: Perhaps a more reasonable question after thinking about the answer I got would have been. Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
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Measurable functions with non measurable image

I am just curious about examples of measurable functions $f:[0,1]\to[0,1]$ such that $f[0,1]$ is not measurable. This is motivated by the question Is measure preserving function almost surjective?, ...
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Change of variables for double integral [closed]

Thank you for your time. My basic question is whether the following change of variables allowed $$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$ I fail to ...
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Does every non-locally compact metric space admit a violation of Lebesgue's theorem?

From the results of Preiss and Tišer, it is known that many natural families of measures on Hilbert spaces violate the Lebesgue Density Theorem. Question: Does every non-locally compact metric space ...
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Sigma algebra of stochastic process

A stochastic process is a collection $(X_t)_{t\in T}$ of random variables from a prob. space $(\Omega,\mathcal{F},P)$ to some measurable space $(E,\mathcal{E})$. Now, in order to understand the whole ...
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Weak$^*$ topology on Bochner $L^p$-spaces

Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...
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Duality of Bochner $L^{\infty}$ space

Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space $$E:=L^{\infty}([0,1],X),$$ i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
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Non-measurability of time integral of non-jointly measurable process

I'm teaching a seminar on probability theory and I want to motivate why joint measurability of a stochastic process is important. The following seems to be the canonical counterexample for a process ...
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Measurably-isomorphic subsets of polish spaces and the continuum hypothesis

In Theorem 2.7 in the following notes, we seem to assume the following statement. Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...
Identity map between $L^2(\mu)$ and $L^2(\mu_{\rm sf})$
Let $(X,\Sigma,\mu)$ be a measure space. The semi-finite version of $\mu$ on $(X,\Sigma)$ is denoted $\mu_{\rm sf}$ and given by  \mu_{\rm sf}(E) = \sup\{\mu(A) \mid A \subseteq E \text{ measurable,...