Taylor Measures in Analysis & Probability

• Taylor measures are rigorous mathematical objects that encode Taylor series coefficients as signed measures, bridging classical analysis, measure theory, and probability.
• They enable the representation of analytic functions through deterministic constructions and facilitate simulation-based estimation in stochastic processes.
• Their Hilbert space and Polish space structures provide a unifying framework for generalizing Taylor expansions, discrete distributions, and process modeling.

Taylor measures encompass a collection of rigorous mathematical constructions—deterministic and stochastic—that encode the coefficients of Taylor series as signed measures, thereby unifying elements of classical analysis, measure theory, and probability. These objects support the transfer of analytic properties, stochastic modeling, and probabilistic measures within a single formal framework. Their recent introduction as articulated in (Micheas, 6 Aug 2025) extends classical Taylor expansions, enables the systematic construction of stochastic processes and discrete probability measures, and establishes a foundation for function encoding in analytic and probabilistic analyses.

1. Deterministic Taylor Measures: Definition and Foundational Properties

Given a sequence $a = [a_0, a_1, a_2, \ldots]$ of real numbers and a real parameter $s$, the deterministic Taylor measure $T_{s,a}$ is the signed measure on the Borel subsets $B \subseteq \mathbb{N}$ defined by

$T_{s,a}(B) = \sum_{n\in B} a_n \frac{s^n}{n!}$

This captures the Taylor expansion coefficients of an analytic function $f$ with $a_n = f^{(n)}(x_0)$ and $s = x-x_0$ via

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n = T_{x-x_0,a}(\mathbb{N})$$

If $a_n \geq 0$, $T_{s,a}$ is a positive measure; in general, it is signed, in which case the Jordan decomposition gives $T_{s,a} = T_{s,a}^+ - T_{s,a}^-$, where $T_{s,a}^\pm$ are the positive/negative variations.

The Radon–Nikodym derivative with respect to the counting measure, $p_T(n) = a_n\, s^n/n!$, provides a canonical density.

Functional-Analytic Structure:

The set of all finite Taylor measures

$$\mathcal{T}^f = \left\{T_{s,a} : T_{s,a}(\mathbb{N}) < +\infty \right\}$$

admits an inner product

$$(T_{s_1,a_1}, T_{s_2,a_2})(B) = \sum_{n \in B} a_{1,n} a_{2,n} \frac{(s_1 s_2)^n}{n!}$$

and induced norm

$\|T_{s,a}\| = \sqrt{(T_{s,a}, T_{s,a})},$

endowing $\mathcal{T}^f$ with a complete separable Hilbert structure (thus a Polish space).

Probability Measures and Function Representation:

Normalizing $T_{s,a}^+$ (for $a_n \geq 0$) yields the "Taylor probability measure"

$$Q_{s,a}^+(B) = \frac{\sum_{n\in B} a_n s^n/n!}{T_{s,a}^+(\mathbb{N})}$$

Any discrete measure absolutely continuous to counting measure is a positive Taylor probability measure. Every analytic function $f$ is representable as a Taylor measure:

$$f(x) = T_{x-x_0,a}(\mathbb{N}), \quad a_n = f^{(n)}(x_0)$$

This yields a function-encoding mechanism via measure theory.

2. Stochastic Taylor Measures and Probabilistic Generalization

Let $(\Omega, A, P)$ be a probability space. A stochastic Taylor measure (STM) is a measurable map

$X : \Omega \to (\mathcal{T}^f, \mathcal{V}),$

where $\mathcal{V}$ renders all $T \mapsto T(B)$ measurable. Explicitly,

$$X(\omega)(B) = \sum_{n\in B} a_n(\omega) \frac{s(\omega)^n}{n!},$$

where $a_n, s$ are random variables.

Illustrative Constructions:

• If $a_n(\omega)$ are independent Gaussian random variables, the mean of $X$ recovers the deterministic Taylor expansion for analytic $f$.
• Setting $a_n(\omega)$ as partial sums of an independent sequence, $X(\omega)(B)$ encodes a random walk.
• Various STM choices yield martingales, indicator functions, and even approximations to Brownian motion—demonstrating the model’s breadth.

The STM formalism supports simulation-based estimation: sample-averages of functions of the form $I(n\in B)$, under the induced densities, approximate expected values of the corresponding subset measures.

3. Relation to Classical Analysis, Probability, and Measure Theory

Taylor measures form a bridge between classical analysis (via Taylor expansions), probability (via discrete distributions), and measure theory (as signed measures of countable additivity).

• Any analytic function can be realized as the total measure of a Taylor measure.
• Discrete probability measures (with absolute continuity to counting measure) are subsumed as Taylor probability measures.
• STM constructions encapsulate a wide variety of stochastic processes, providing a unified mathematical perspective.

This synthesis extends to simulation and state-space modeling (autoregressive, random walks, Brownian motion), as well as to function approximation, spatial statistics, and differential equations.

4. Framework for Analytic, Discrete, and Stochastic Objects

The Taylor measure structure supports the following operations and properties:

Operation	Deterministic	Stochastic
Measure Definition	$T_{s,a}(B)$	$X(\omega)(B)$
Jordan Decomposition	$T_{s,a}=T^+-T^-$	$X(\omega)=X^+(\omega)-X^-(\omega)$
Inner Product & Norm	$(T_{s_1,a_1},T_{s_2,a_2})$	$\mathbb{E}[(X_1,X_2)]$
Representation of Function $f$	$f(x)=T_{x-x_0,a}(\mathbb{N})$	$\mathbb{E}[X(\mathbb{N})]$
Simulation/Approximation	None explicit	Via sample averages

A plausible implication is that the Taylor measure approach provides new analytic and computational tools for encoding and manipulating analytic and stochastic quantities in a measure-theoretic language, while retaining structure amenable to both probabilistic and analytic manipulation.

5. Connections to Measure-Theoretic and Probabilistic Constructions

The Taylor measures’ construction leverages:

• Radon–Nikodym theory: For expressing densities (Taylor derivatives) with respect to counting or other base measures.
• Jordan decomposition: For handling the signed nature of general Taylor measures and isolating positive and negative parts for normalization.
• Polish space structure: Via explicit inner products and countable dense subsets—crucial for measure-theoretic completeness and measurable selection theorems.
• State-space and random process modeling: By embedding, via the sequence $a_n(\omega)$, structures such as Markov chains, random walks, and even functionals of Gaussian processes.

6. Summary and Broader Mathematical Implications

Taylor measures, both deterministic and stochastic, offer a unified framework that generalizes Taylor expansions to the field of signed measures on $\mathbb{N}$ and their randomizations. This yields:

• A measure-theoretic encoding for analytic functions, where entire function classes are in bijection with measure spaces defined by their Taylor coefficients.
• Construction of discrete probability distributions and associated processes as normalized Taylor measures.
• Encoding of stochastic processes, martingales, and simulation frameworks as instances of stochastic Taylor measures.
• New Hilbert- and Polish-space structures for spaces of measures encoding analytic and probabilistic information.

Through canonical identities, inner-product structures, and probabilistic extensions, Taylor measures generalize and connect objects across functional analysis, probability, measure theory, and numerical approximation, providing a fertile setting for further algorithmic and theoretical development (Micheas, 6 Aug 2025).