  In this paper, we study norm almost periodic measures on locally compact
Abelian groups. First, we show that the norm almost periodicity of $\mu$ is
equivalent to the equi-Bohr almost periodicity of $\mu*g$ for all $g$ in a
fixed family of functions. Then, we show that, for absolutely continuous
measures, norm almost periodicity is equivalent to the Stepanov almost
periodicity of the Radon--Nikodym density.
