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The concept of dimension is an important task in geometry. It permits a description of the growth process of objects. It may be seen as an invariant measure characterizing the object. Fractal dimensions are a kind of invariants permitting essentially to describe the irregularity hidden in irregular objects, by providing a suitable growth law. Among fractal geometrical objects, Moran’s types play an important role in explaining many situations, in pure mathematics as the general context of Cantor’s, and in applied physics as a suitable context for studying scaling laws. In the present paper, some non-regular homogeneous Moran measures are investigated, by establishing some new sufficient conditions permitting an explicit computation of the relative multifractal dimensions of the level sets for which the classical formulation does not hold. Besides, the mutual singularity of the relative multifractal measures for the homogeneous Moran case with different multifractal dimensions is investigated. This is very important, as in quasi-all existing situations, the validity of the multifractal formalism passes through the equality of the multifractal Hausdorff dimension with the packing one.

In this paper, we attain the multifractal Hewitt–Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt–Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples.

M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

We propose an example for which the multifractal Hausdorff and packing measures are mutually singular.

The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.