Lebesgue integrals containing a parameter, with applications

The Riemann and Lebesgue Integrals

§1 Preliminaries: Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §2 Riemann Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 §3 Lebesgue measure zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 §4 Definition and Properties of the Lebesgue Integral . . . . . . . . . . 7 §5 The spaces L(R) and L(R) ....

Feynman parameter integrals

We often deal with products of many propagator factors in loop integrals. The trick is to combine many propagators into a single fraction so that the four-momentum integration can be done easily. This is done commonly using so-called Feynman parameters. We rewrite the product of propagators 1 (A 1 + ii)(A 2 + ii) · · · (A n + ii) , (1) where A i has the form of p 2 − m 2. The sign of A i is not...

Multi-parameter singular integrals

On the Substitution Rule for Lebesgue–stieltjes Integrals

We show how two change-of-variables formulæ for Lebesgue–Stieltjes integrals generalize when all continuity hypotheses on the integrators are dropped. We find that a sort of “mass splitting phenomenon” arises. Let M : [a, b]→ R be increasing. Then the measure corresponding to M may be defined to be the unique Borel measure μ on [a, b] such that for each continuous function f : [a, b] → R, the i...

A NEW FOUR-PARAMETER DISTRIBUTION: PROPERTIES AND APPLICATIONS

In this paper a new four-parameter lifetime distribution named “the Exponentiated gompertz-poisson (E-GP) distribution” has been suggested that it has a decreasing, increasing, bathtub-shaped and inverse bathtub-shape for modeling lifetime data. The Exponentiated gompertz-poisson has applications in economics, actuarial modelling,reliability modeling, lifetime and queuing problems and biologica...