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This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let $(\Omega ,ℱ,\mathbb{P})$ be a probability space and $p(·):\Omega \to (0,\infty )$ be a $ℱ$-measurable function such that $0<{inf}_{x\in \Omega }p\left(x\right)\le {sup}_{x\in \Omega }p\left(x\right)<\infty$. It is proved that a predictable martingale Hardy space ${𝒫}_{p(·)}$ has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with...