In this note we give an overview of recent results in the theory of electrorheological fluids and the theory of function spaces with variable exponents. Moreover, we present a detailed and self-contained exposition of shifted $N$-functions that are used in the studies of generalized Newtonian fluids and problems with $p$-structure.

Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m,1}$, $m\ge 1$. Let $E\subset X$ be a set of measure zero. Then ${\widehat{\mathscr{H}}}_m(E\cap u^{-1}\left(y\right))=0$ for ${\mathscr{H}}_m$-a.e. $y\in Y$, where ${\mathscr{H}}_m$ is the $m$-dimensional Hausdorff measure and ${\widehat{\mathscr{H}}}_m$ is the $m$-codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of...