Minkowski addition and subtraction of functions and sets made insensitive to null sets

The two operations of Minkowski addition and subtraction, defined both for sets and for functions, have found many applications in integral geometry, morphological image processing and spatial logic, generally within the digital framework of subsets of $\mathbb{Z}^n$ and functions $\mathbb{Z}^n\to\mathbb{Z}$. However, in the Euclidean framework of subsets of $\mathbb{R}^n$ and functions $\mathbb{R}^n\to\mathbb{R}\cup\{-\infty,+\infty\}$, they suffer from a defect: applying them to two sets or functions that differ only in a set of zero Lebesgue measure can lead to fundamentally different results. In this paper, we remedy to this problem. Our framework is the complete lattice of equivalence classes of Borel functions under equality almost everywhere. We replace the usual operations of numerical supremum and infimum by their essential forms with respect to a $\sigma$-finite Borel measure applied to the second argument of the addition or subtraction, called the ``structuring element''; we examine two particular cases, the Lebesgue measure and the discrete measure for a countable structuring element. The algebraic and lattice-theoretical properties of the Minkowski operations are preserved in this framework. For the sake of completeness, we recall in detail the lattice-theoretical background of our theory and give the precise formalism for Minkowski operations on functions $\mathbb{R}^n\to\mathbb{R}\cup\{-\infty,+\infty\}$.