We investigate the connections between different types of continuity of the null-additive fuzzy measure $m$ (monotone set function which vanishes at the empty set and such that $m(B)=0$ implies $m(A\cup B)=m(A)$) as continuity from above, continuity from below, autocontinuity from above, autocontinuity from below, continuity with respect to other fuzzy measure. A Lebesgue type decomposition theorem for continuous and autocontinuous from above fuzzy measure is proved.