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In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak type and strong type inequalities on Doob's maximal operator and get a $(1,p(\cdot),\infty)$-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John-Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with probabilistic characterization of $p(\cdot)$ to replace the so-called log-H\"{o}lder continuity condition in $\mathbb {R}^n.$Comment: Banach Journal of Mathematical Analysis, to appea

Let p(·) be a measurable function on double-struckRn with 0

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