# Questions tagged [measure-theory]

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

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### Kolmogorov entropy of a subset of $L^1$

How can we estimate the Kolmogorov $\epsilon$-entropy $$H_\epsilon (A,L^1(\mathbb R))$$ where $A = \{f:\mathbb R \to [0,K] \text{ s.t.$f \in L^1$and has total variation$TV(f) \le M$}\}$?
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### Example of a strictly proper scoring rule defined on the set of all probability measures on $[0,1]$

This question is closely related to another question I asked recently but is more to the point than that other question. Let $\mathcal P$ be the set of all probability measures on the Borel algebra of ...
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### Was Cantor aware of Lebesgue theory of integration? [closed]

Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
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### Example of a (strictly) proper scoring rule on a general measurable space?

Most of the literature on scoring rules that I know of deals with discrete measurable spaces, but in this paper by Gneiting and Raferty a very general definition of a scoring rule is given. I don't ...
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Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \... 0answers 30 views ### Binary law on pairs of finite unions of segments Let$U$be the set of all nonempty subsets of$[0,1]$that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ... 1answer 85 views ### To show a set is a set of positive Lebesgue measure in$ \mathbb{R}$Let$E\subset \mathbb{R}$be a set of positive Lebesgue measure. Can we find$l>0$such that $$\bigcap_{-l\leq t \leq l}t+E$$ is a set of positive Lebesgue measure? Notation:$t+E=\{t+e|e\in E\}$1answer 73 views ### Measurability of superposition operator with non-separable Banach space Let$f\colon I \times X \to \mathbb{R}$be a map where$I \subset \mathbb{R}$is an interval,$X$is a Banach space (possibly non-separable) and we have $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \... 1answer 89 views ### Spectrum of a self-adjoint operator and spectral measures Let T be a self-adjoint operator on a Hilbert space \mathcal{H}, with spectrum \sigma(T). For any x,y\in \mathcal{H}, denote by \mu_{xy} the spectral measure of T with respect to x and ... 0answers 71 views ### Recursive expression of Lebesgue measure for balls with removed intersection This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known). Let \left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N be a set ... 2answers 101 views ### Induced probability measure on a finite orbit under a group action Suppose we have a discrete group G acting on a compact set X \subseteq \mathbb{R}^d via measure-preserving homeomorphisms, and suppose we have a point x whose orbit Gx is finite (say |Gx| = n... 1answer 212 views ### Regarding a positive Lebesgue measure set in \mathbb{R}^2 Let P\subset \mathbb{R}^2 be a positive Lebesgue measure set. Then P does not necessarily contain a subset of the form A\times B where A,B\subset \mathbb{R} are of positive Lebesgue measure. ... 1answer 54 views ### Convergence in weak dual topology \sigma(L^\infty, L^1) Let f\in L^\infty(\mathbb{R})\cap C(\mathbb{R}), that is f is continuous and bounded on \mathbb{R}. Let S_r denote the shift by r\in \mathbb{R}: S_r f=f(\cdot-r). Suppose S_{r} f ... 0answers 53 views ### How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field? I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also. Preliminaries An algebra of sets in a set X is an \mathcal{X}\subseteq\mathcal{P}... 1answer 166 views ### Is there a standard way of defining the integral of an extended real function with respect to a finitely additive probability measure? Let X be a set, and let \mu be a finitely additive probability measure defined on 2^X. Let \Phi be the set of functions from X to \mathbb R \cup \{-\infty, \infty\}. Is there a standard ... 1answer 257 views ### Do Borel subsets of the plane with null sections have Borel projections? This might be a very easy question, and it might be better for mathstackexchange in which case I apologize. I'm stuck on something an anonymous referee wrote to me about a paper of mine and I'm hoping ... 1answer 104 views ### Unique solution of a 1-D ODE with a bounded positive right-hand-side Consider the initial value problem$$\dot x(t) = F(t,x), \quad t \in (0,T)$$with given initial datum$$x(0) = x_0 \in \mathbb R.$$More precisely we consider the integral equation$$x(t)=x(0)+\int_0^... 1answer 95 views ### Interpolation inequality$\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$[closed] Let$u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds? $$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$ 0answers 59 views ### Covering R2 by rectifiable curves and null sets We have a family indexed by$t \in \mathbb R$of rectifiable Jordan curves$(\gamma_t)$, such that$\bigcup_{t\in \mathbb R}\gamma_t = \mathbb R^2 \setminus \{0\}$. Moreover, the family is monotone, ... 1answer 56 views ### Compute limit of$\mathbb P(Y \le X_n)$using limiting information on the sequence of random variables$X_n$Let$Y$be a symmetric random variable,$(X_n)_n$be a sequence of nonnegative random variables, and set$p_n = \mathbb P(Y \le X_n)$. It is known from Slutsky's theorem that, if$c$is a constant ... 1answer 124 views ### Integration theory for functions and values with values in topological rings I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings. The generalization of a measure ... 0answers 40 views ### Entropy of flow and time-1 map Let$\Phi=(\phi_t)_{t\in \mathbb{R}}$be a continuous flow on a compact metric space$X$. Let$\mu$be a$\phi_1$-invariant measure. Then it is not hard to verify tht$\int_{0}^{1} \phi_t\mu dt$is$\...
By sum of two sets I mean $A+B := \{x+y:x \in A \quad y \in B\}$, and there is a tip in a book of real analysis by Zhou Minqiang which says: “If $A,B$ are Borel sets in $\mathbb{R}^{n}$, $A+B$ may not ...