Bicomplex Version of Lebesgue's Dominated Convergence Theorem and Hyperbolic Invariant Measure

TL;DR

This paper extends fundamental measure theory results, including Lebesgue's dominated convergence theorem and Lebesgue-Radon-Nikodym theorem, to bicomplex valued functions and introduces a hyperbolic invariant measure concept.

Contribution

It develops bicomplex measure theory by proving bicomplex versions of key theorems and introduces a novel hyperbolic invariant measure concept.

Findings

Bicomplex Lebesgue dominated convergence theorem established

Bicomplex Lebesgue-Radon-Nikodym theorem proved

Introduction of hyperbolic invariant measure

Abstract

In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue's dominated convergence theorem and some other results related to this theorem. Also we have proved the bicomplex version of Lebesgue-Radon-Nikodym theorem. Finally we have introduced the idea of hyperbolic version of invariant measure.