arXiv subjects
Publications and source records attributed to Peter A. Loeb.
Previously, the authors used the insights of Robinson's non-standard analysis as a powerful tool to extend and simplify the construction of some compactifications of regular spaces. They now show that any Hausdorff compactification is obtainable with their method.
An intuitive probabilistic alternative for the construction of the Martin boundary is presented along with a construction of maximal representing measures for positive harmonic functions.
It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a preassigned $ε$ of the integral, with the sum for the local errors also less than $ε$. All of this follows from the ubiquity of Lebesgue points, which is a consequence of Lusin's theorem, for which a simple proof is included in the discussion.
This paper shows how the Lebesgue integral can be obtained as a Riemann sum and provides an extension of the Morse Covering Theorem to open sets. Let $X$ be a finite dimensional normed space; let $μ$ be a Radon measure on $X$ and let $Ω\subseteq X$ be a $μ$-measurable set. For $λ\geq1$, a $μ$-measurable set $S_λ(a)\subseteq X$ is a $λ$-Morse set with tag $a\in S_λ(a)$ if there is $r>0$ such that $B(a,r)\subseteq S_{λ}(a)\subseteq B(a,λr)$ and $S_λ(a)$ is starlike with respect to all points in the closed ball $B(a,r)$. Given a gauge $δ:Ω\to(0,1]$ we say $S_λ(a)$ is $δ$-fine if $B(a,λr)\subseteq B(a,δ(a))$. If $f\geq0$ is a $μ$-measurable function on $Ω$ then $\int_Ωf dμ=F\in\mathbb{R}$ if and only if for some $λ\geq1$ and all $ε>0$ there is a gauge function $δ$ so that $|\sum_{n}f(x_{n}) μ(S(x_{n}))-F|<ε$ for all sequences of disjoint $λ$-Morse sets that are $δ$-fine and cover all but a $μ$-null subset of $Ω$. This procedure can be applied separately to the positive and negative parts of a real-valued function on $Ω$. The covering condition $μ(Ω\setminus\cup_{n}S(x_{n}))=0$ can be satisfied due to the Morse Covering Theorem. The improved version given here says that for a fixed $λ\geq1$, if $A$ is the set of centers of a family of $λ$-Morse sets then $A$ can be covered with the interiors of sets from at most $κ$ pairwise disjoint subfamilies of the original family; an estimate for $κ$ is given in terms of $λ$, $X$ and its norm.