Separately σ-additive and separately finitely additive complex functions on the Cartesian product of two algebras of sets are represented in terms of spectral measures and their finitely additive counterparts. Applications of the techniques include a bounded joint convergence theorem for bimeasure integration, characterizations of positive-definite bimeasures, and a theorem on decomposing a bimeasure into a linear combination of positive-definite ones.

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems...

We study Lp(Rn) → Ldμ(σ)α,∞(Ldt∞) estimates for the Radon transform in certain cases where the dimension of the measure μ on Σ(n-1) is less than n-1.

We first provide an approach to the conjecture of Bierstone-Milman-Pawłucki on Whitney’s problem on $C^d$ extendability of functions. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on flatness.

We show that a comeager Π₁¹ hereditary family of compact sets must have a dense $G_{\delta }$ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true $F_{\sigma \delta }$ sets.

By iterating the Bolyai-Rényi transformation $T\left(x\right)={(x+1)}^2(mod1)$, almost every real number $x\in [0,1)$ can be expanded as a continued radical expression $$x=-1+\sqrt{x_1+\sqrt{x_2+\cdots +\sqrt{x_n+\cdots }}}$$ with digits $x_n\in \{0,1,2\}$ for all $n\in \mathbb{N}$. For any real number $x\in [0,1)$ and digit $i\in \{0,1,2\}$, let $r_n(x,i)$ be the maximal length of consecutive $i$’s in the first $n$ digits of the Bolyai-Rényi expansion of $x$. We study the asymptotic behavior of the run-length function $r_n(x,i)$. We prove that for any digit $i\in \{0,1,2\}$, the Lebesgue measure of the set $$D\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\frac{1}{log{\theta }_i}\right\}$$ is $1$, where ${\theta }_i=1+\sqrt{4i+1}$. We also obtain that the level set $$E_{\alpha }\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\alpha \right\}$$ is of full Hausdorff dimension...