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Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, T, in [Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson, Convergence in Riesz spaces with conditional…

The action and averaging properties of conditional expectation operators are studied in the, measure-free, Riesz space, setting of Kuo, Labuschagne and Watson [{Conditional expectations on Riesz spaces}, J. Math. Anal. Appl., 303 (2005),…

Let E be a Dedekind complete Riesz space with weak unit e, equipped with a conditional expectation operator T. We prove that the spaces Lp(T), with their natural vector-valued norms, are strongly complete, extending the p=2 case of Kuo,…

This note generalizes the notion of conditional probability to Riesz spaces using the order-theoretic approach. With the aid of this concept, we establish the law of total probability and Bayes' theorem in Riesz spaces; we also prove an…

We introduce a novel real-valued endogenous logic for expressing properties of probabilistic transition systems called Riesz modal logic. The design of the syntax and semantics of this logic is directly inspired by the theory of Riesz…

In the recent surge of papers on ergodic theory within Riesz spaces, this article contributes by introducing enhanced characterizations of ergodicity. Our work extends and strengthens prior results from both the authors and Homann, Kuo, and…

We use integration by parts formulas to give estimates for the $L^p$ norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and…

We extend the concept of conditional supremum to the measure-free setting of Riesz spaces via the conditional expectation operator. We explore its properties and show how this tool is crucial in generalizing various results across multiple…

It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space $M^\infty_{R}\big(\sum_k T_k\big)$ of Riesz summability…

Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincar\'{e} Recurrence Theorem and the Kac…

In this paper we generalize the H\'ajek-R\'enyi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of…

For a conditional expectation operator $T$ on a Dedekind complete Riesz space, we give representations of the $T$-strong duals of $L^1(T)$ and $L^\infty(T)$. The representation for the $T$-strong dual of $L^1(T)$ follows from the known…

In [Math. Ineq. \& appl., Vol 26 (2) (2023), 511-530] and [Period. Math. Hung., 89 (1) (2024), 116-128], the present author proved that the Riesz potential $I_{\alpha}$ extends to a bounded operator $H^{p(\cdot)}_{\omega}(\mathbb{R}^n) \to…

We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [{{\sc B. A. Watson}, {An And\^o-Douglas type theorem in Riesz spaces with a conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a…

We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index $(d-1)/2$ may fail for functions in the Hardy space $h^1(\mathbb T^d)$, we prove…

This work develops, from a functional analytic perspective, the construction of random variables in Lebesgue spaces L^p. It extends classical notions of measurability, integrability, and expectation to L^p valued functions, using Pettis's…

\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the…

The continuity of conditional expectation on Orlicz spaces is investigated. Indeed, we provide some necessary and sufficient conditions on a sequence $\{\mathcal{A}_n\}_{n\in\mathbb{N}}$ of $\sigma$-subalgebras for $L^{\varphi}$-convergence…

We study orthogonally additive operators between Riesz spaces without the Dedekind completeness assumption on the range space. Our first result gives necessary and sufficient conditions on a pair of Riesz spaces $(E,F)$ for which every…

For $2\le p<\infty$ we show the lower estimates \[ \|A^{\frac 12}x\|_p \kl c(p)\max\{\pl \|\Gamma(x,x)^{{1/2}}\|_p,\pl \|\Gamma(x^*,x^*)^{{1/2}}\|_p\} \] for the Riesz transform associated to a semigroup $(T_t)$ of completely positive maps…