Integration Recall that given a measure space (Ω,F , μ), we denote by L = L(Ω,F , μ) the vector space of integrable functions. Besides defining integrable functions, we call a function f integrable on F in F if 1F f is integrable and we write ∫

A k-fan in the plane is a point x ∈R2 and k halflines starting from x. There are k angular sectors σ1, . . . , σk between consecutive halflines. The k-fan is convex if every sector is convex. A (nice) probability measure μ is equipartitioned by the k-fan if μ(σi) = 1/k for every sector. One of our results: Given a nice probability measure μ and a continuous function f defined on sectors, there ...

Let M be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension d ≥ 1. Let m be some (smooth) volume probability measure in M. Let C(M,M) be the set of C maps M → M, endowed with the C topology. Given f ∈ C(M,M), we say that μ is an acim for f if μ is an f -invariant probability measure which is absolutely continuous with respect to m. Theorem 1. The set R of C map...

The integration with respect to a vector measure may be applied in order to approximate a function in a Hilbert space by means of a finite orthogonal sequence {fi} attending to two different error criterions. In particular, if ∈ R is a Lebesgue measurable set, f ∈ L2( ), and {Ai} is a finite family of disjoint subsets of , we can obtain a measure μ0 and an approximation f0 satisfying the follow...

20.1. Additional technical results on weak convergence Given two metric spaces S1, S2 and a measurable function f : S1 → S2, suppose S1 is equipped with some probability measure P. This induces a probability measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf −1 [X] is equal to ...

It is known that if the support of a function $f \in L^{1}(\mathbb{R}^{n})$ and its Fourier transform have finite measure then $f=0$ almost everywhere. We study generalizations this property for semisimple Lie groups.

An equivalence between the Gurov-Reshetnyak GR(ε) and Muckenhoupt A∞ conditions is established. Our proof is extremely simple and works for arbitrary absolutely continuous measures. Throughout the paper, μ will be a positive measure on R absolutely continuous with respect to Lebesgue measure. Denote Ωμ(f ;Q) = 1 μ(Q) ∫ Q |f(x) − fQ,μ|dμ(x), fQ,μ = 1 μ(Q) ∫

Let f : z 7→ z + c be a quadratic polynomial whose Julia set J is locallyconnected. We prove that the Brolin measure of the set of biaccessible points in J is zero except when f(z) = z − 2 is the Chebyshev quadratic polynomial for which the corresponding measure is one. §

Let K be a compact set of Rn and t ? 0. In this paper, we discuss the relation between t-dimensional Hewitt-Stromberg premeasure measure denoted by H?t Ht respectively. We prove : if (K) < +? then = Ht(K) +?, there exists subset F such that (F) Ht(F) is close as like to Ht(K).

This study aims to 1) develop Machine Learning Ecosystem models enhance grade point averages, and 2) predict averages by modeling techniques, namely, Decision Trees (DT), Naïve Bayes (NB), a Neural Network (NN). Findings from an efficiency comparison of the three in predicting showed that DT achieved highest accuracy 100.00%. In contrast, NN second-highest 85.83%, NB lowest 81.67%. For F-Measur...