, and otherwise. If the delta function is conceptualized as modeling an idealized point mass at 0, then represents the mass contained in the set . One may then define the integral against as the integral of a function against this mass distribution. Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure satisfies for all continuous compactly supported functions . The measure is not absolutely continuous with respect to the Lebesgue measure—in fact, it is a singular measure. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property holds. As a result, the latter notation is a convenient abuse of notation, and not a standard (Riemann or Lebesgue) integral. As a probability measure on , the delta measure is characterized by its cumulative distribution function, which is the unit step function. This means that is the integral of the cumulative indicator function with respect to the measure ; to wit, the latter being the measure of this interval; more formally, . Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral: All higher moments of are zero. In particular, characteristic function and moment generating function are both equal to one. As a distribution In the theory of distributions, a generalized function is considered not a function in itself but only about how it affects other functions when "integrated" against them. In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" test function φ. Test functions are also known as bump functions. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral. A typical space of test functions consists of all smooth functions on R with compact support that have as many derivatives as required. As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by for every test function . For δ to be properly a distribution, it must be continuous in a suitable topology on the space of test functions. In general, for a linear functional S on the space of test functions to define a distribution, it is