We construct a family of measures called infinite products which generalize Gibbs measures in the one-dimensional lattice gas model. The multifractal properties of these measures are stndied under some regularity conditions. In particular, if the r-function is differentiable, we prove a formula which gives the Hausdorff dimension and packing dimension of the set of singularity points of a given order. Mathematical examples include Riesz products, g-measures, and G-measures.