  Let $S_{n}f$ be the $n$th partial sum of the Vilenkin-Fourier series of $f\in
L^{1}(G).$ For $1<p_{-}\leq p_{+}<\infty,$ we characterize all exponent
$p(\cdot)$ such that if $f\in L^{p(\cdot)}(G)$, $S_{n}f$ converges to $f$ in
$L^{p(\cdot)}(G).$
