  We investigate the properties of the variable Lebesgue spaces with quasi-norm
on a probability space, and give the atomic decompositions suited to the
variable exponent martingale Hardy spaces. Using the decompositions and the
harmonic mean of a variable exponent, we obtain several continuous embedding
relations between martingale Hardy spaces with small exponent. Finally, we
extend these results to the cases $0<p^-\leq p^+<\infty.$
