necessary and sufficient that, for every positive integer N there is an integer MN and a constant CN such that for every test function φ, one has the inequality where sup represents the supremum. With the δ distribution, one has such an inequality (with with for all N. Thus δ is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being {0}). The delta distribution can also be defined in several equivalent ways. For instance, it is the distributional derivative of the Heaviside step function. This means that for every test function φ, one has Intuitively, if integration by parts were permitted, then the latter integral should simplify to and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation () defines a Daniell integral on the space of all compactly supported continuous functions φ which, by the Riesz representation theorem, can be represented as the Lebesgue integral of φ concerning some Radon measure. Generally, when the term "Dirac delta function" is used, it is in the sense of distributions rather than measures, the Dirac measure being among several terms for the corresponding notion in measure theory. Some sources may also use the term Dirac delta distribution. Generalizations The delta function can be defined in n-dimensional Euclidean space Rn as the measure such that for every compactly supported continuous function f. As a measure, the n-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with , one has The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case. However, despite widespread use in engineering contexts, () should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances. The notion of a Dirac measure makes sense on any set. Thus if X is a set, is a marked point, and Σ is any sigma algebra of subsets of X, then the measure defined on sets by is the delta measure or unit mass concentrated at x0. Another common generalization of the delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the