These notes record the lectures on Lebesgue integration, which is a topic not covered in the textbook. It is largely inspired by the approach of Leon Simon when he previously taught this course. (All mistakes are my own of course!) The notes will be posted after each lecture. Any comments or corrections, even very minor ones, are very much appreciated! Our goal is to define the notion of Lebesg...

this work deals with the calculation of franck-condon factors using hua potential function for the first time. the advantages of this function have been mentioned, and the numerical methods are used to obtain franck-condon factors for the following band systems: agreement between the values of this work and those obtained from other works is quite good that verifies the reliability of the prese...

We calculate the Lebesgue–measures of the stability sets of Nashequilibria in pure coordination games. The results allow us to observe that the ordering induced by the Lebesgue–measure of stability sets upon strict Nashequilibria does not necessarily agree with the ordering induced by risk–dominance. Accordingly, an equilibrium selection theory based on the Lebesgue–measure of stability sets wo...

In the context of Lebesgue integration, we derive the divergence theorem for unbounded vector fields that can have singularities at every point of a compact set whose Minkowski content of codimension greater than two is finite. The resulting integration by parts theorem is applied to removable sets of holomorphic and harmonic functions. In the context of Lebesgue integration, powerful divergenc...

In this paper, we introduce the p-adic Hardy type operator and obtain its sharp bound on the p-adic Lebesgue product spaces. Meanwhile, an analogous result is computed for the p-adic Lebesgue product spaces with power weights. In addition, we characterize a sufficient and necessary condition which ensures that the weighted p-adic Hardy type operator is bounded on the p-adic Lebesgue product spa...

An algorithm is presented to compute good point sets and weights for discrete least squares polynomial approximation on a geometry Ω ⊂ R2. The criterion that is used is the minimisation of the Lebesgue constant of the corresponding least squares operator. In order to approximate the Lebesgue constant, we evaluate the Lebesgue function in a point set generated by a refinement method that is base...

In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to non-Gaussian) fractional processes from a finite sum Wiener chaoses is treated. The family that considered includes, for example, Brownian motions any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. class includes large variety most commonly used functio...

We prove optimal integrability results for solutions of the p(·)-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials map L to variable exponent weak Lebesgue spaces.