Multiplicative Analysis

Chapter 1: It begins with assumptions made on properties of real numbers. The classical additive modulus function along with the corresponding additive metric are defined. The multiplicative modulus function with a corresponding multiplicative metric are defined. Properties for both types are presented, but proofs for properties for usual additive modulus function are not presented. Topologies for both functions have been presented without providing definitions for general topologies, general metrics and general multiplicative metrics. Chapter 2: The concept of topology and properties of topology are discussed. Only results which are essential for the development of infinite products and multiplicative measure integration are discussed. However, continuity, compactness, local compactness, and connectedness have been discussed. Convergences in terms of nets are discussed. Chapter 3: Classical additive metrics are not presented, but multiplicative metrics are introduced. Balls, topologies, and topological properties including totally boundedness and uniform continuity derivable from multiplicative metrics have been discussed. Fundamental fixed point theorems have been derived. Chapter 4: Convergence of infinite products of positive real numbers has been defined by using multiplicative modulus function, which is equivalent to the classical definition. It begins with the result which states that every multiplicative absolute convergent infinite product converges. Here, multiplicative absolute convergence is defined with the help of multiplicative modulus function; which is different from usual absolute convergence of infinite products. It has been established that multiplicative modulus function is the best tool to discuss the theory of infinite products. Unordered convergence has also been discussed for infinite products. Riemann rearrangement theorem for infinite products has also been established with the help of multiplicative modulus function. Chapter 5: The definitions for classical differentiation and classical Riemann integration have not been presented. New definitions for multiplication oriented differentiation and Riemann integration have been presented by using multiplicative modulus function. The first observation is that differentiability implies continuity for this new differentiation. Derivative of an integration coincides with integrand. This result has been established for new multiplication oriented concepts. Chapter 6: Sigma algebra, measurable sets, measurable spaces, and measurable functions. All these classical concepts have been defined, and their fundamental properties have been derived. More specifically, approximation for a non negative measurable function by means of a sequence of simple measurable function has been derived. All preliminary works for the next chapter have been done. Chapter 7: Multiplicative lengths for intervals and Lebesgue multiplicative (outer) measure in the set of positive real numbers have been introduced. Classical Lebesgue measurable sets have been defined Multiplication based abstract multiplicative measure has been defined on a classical measurable space. The corresponding important theorems like monotone convergence theorem and Riesz representation theorem have been derived. Second derivation for Lebesgue multiplicative measure through Riesz representation theorem has been presented. Chapter 8: Since Chapter 3 provides only multiplicative metrics, the other concept of multiplicative pseudo metrics is introduced in Chapter 8. Classical uniform spaces are defined without deriving topological spaces directly from uniform spaces. It has been observed that each uniform space provides a family of multiplicative pseudo metrics and that a family of multiplicative pseudo metrics provides a uniformity. More specifically the following multiplicative metrization theorem has been derived. A uniform space is multiplicative metrizable if and only if the uniformity has a countable base.