Radon-Nikodym derivative as a limit of ratios

This question is related to Radon-Nikodym derivatives as limits of ratios. Let $F$, $G$ be sigma-finite measures (or at least probability measures) on $\mathbb{R}$ such that $F \ll G$. The theorem quoted in the link tells that the Radon-Nikodym derivative checks $$\frac{\mathrm{d}F}{\mathrm{d}G} (x) = \lim_{h\to 0^+} \frac{F(x-h, x+h)}{G(x-h, x+h)}$$ for $G$-almost every $x$. Do we … Read more

A strongly open set which is not measurable in the weak operator topology

Let H be a non-separable Hilbert space and {ei}i∈I be an orthonormal basis for H. Let J be a uncountable proper subset in I. Let us put E={x∈B(H):‖ One may check that E is an open set in the strong operator topology but not in the weak operator topology. Question1: I feel E is not … Read more

Topology on the space of Borel measures

Let B be the set of all measures ϕ of Rn such that every open set is ϕ-measurable (sometimes these measures are called Borel measures). Note the measures in B are not required to be locally finite. Let 1≤m≤n be an integer. The upper density of a measure ϕ at a∈Rn is defined as Θ∗m(ϕ,a)=lim sup … Read more

Continuous doubling weight vanishing on set of positive measure?

If I is a bounded interval in R, let 2I denote an interval with the same center point but double the length. A doubling measure on R is a (non-trivial, locally finite, Borel) measure μ such that μ(2I)≤Cμ(I) for all intervals I and some fixed constant C. A doubling weight is an L1loc function w:R→R≥0 … Read more

Approximation of monotone Sobolev functions

Let f∈W1,2loc(R2) be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of f in a precompact open set are attained at the boundary). Is it true that there exists a sequence of smooth monotone functions fn converging to f in W1,2loc(R2)? We could ask the same question for … Read more

Dual of the space of all bounded functions, $B(X, \mathbb{R}).$

Let $X$ be a non compact separable metric space. Denote by $B(X, \mathbb{R})$ the set of all bounded real functions endowed with the sup norm, this is a Banach space. Denote by $C_b(X,\mathbb{R})\subset B(X, \mathbb{R})$ the subspace of all continuous functions, which with the induced norm is also a Banach space. Question 1.: What are … Read more

Lipschitz kernel

We consider the following probability measure on $\mathbb{R}^2$: $\mu = Leb\vert_{[0,1]} \times \delta_0$. Furthermore the following dilation, say $d$, is defined as $(x,0) \mapsto \frac{1}{2}(\delta_{(x,f(x))}+\delta_{(x,-f(x))})$ where $f$ is a non-continuous function function. Define the probability measure $\nu$ as $\nu = \mu d$. Why is this kernel/dilation not a Lipschitz kernel? Short remark about Lipschitz kernel: … Read more

Are there any nontrivial examples of C1C^1 hypersurfaces with bounded (integrable) generalized mean curvature?

The definition of generalized mean curvature on C1 hypersurfaces is given as follows: Let M be a closed orientable C1 hypersurface in Rn+1 and μ be the n-dimensional Hausdorff measure. We say the Rn+1 valued vector function H is a generalized mean curvature on M, if for any smooth vector fields X∈Rn+1, the following identity … Read more

Does it make sense to regard the graph of any function as being a “sort-of-null set”?

Following the nice answer to Do the Lebesgue-null sets cover “all the sets can naturally be regarded as sort-of-null sets”?, the particular situation that I am especially interested in (which is a kind of “advanced version” of https://math.stackexchange.com/questions/35606/lebesgue-measure-of-the-graph-of-a-function) is: Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \times [0,1] \to [0,1] … Read more

Does rate of convergence in probability come from a metric?

In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is from the limit point. This can be done in a metric space as follows: Definition 1 (Rate … Read more