differentiable structure. The delta function on a manifold M centered at the point is defined as the following distribution: for all compactly supported smooth real-valued functions φ on M. A common special case of this construction is a case in which M is an open set in the Euclidean space Rn. On a locally compact Hausdorff space X, the Dirac delta measure concentrated at a point x is the Radon measure associated with the Daniell integral () on compactly supported continuous functions φ. At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping is a continuous embedding of X into the space of finite Radon measures on X, equipped with its vague topology. Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X. Properties Scaling and symmetry The delta function satisfies the following scaling property for a non-zero scalar α: and so Scaling property proof: In this proof, the delta function representation as the limit of the sequence of zero-centered normal distributions is used. This proof can be made by using other delta function representations as the limits of sequences of functions, as long as these are even functions. In particular, the delta function is an even distribution (symmetry), in the sense that which is homogeneous of degree −1. Algebraic properties The distributional product of δ with x is equal to zero: Conversely, if , where f and g are distributions, then for some constant c. Translation The integral of the time-delayed Dirac delta is This is sometimes referred to as the sifting property or the sampling property. The delta function is said to "sift out" the value of f(t) at t = T. It follows that the effect of convolving a function f(t) with the time-delayed Dirac delta is to time-delay f(t) by the same amount. This is sometimes referred to as the shifting property (not to be confused with the sifting property): Note that the sifting property finds the value of a function centered at T whereas the shifting property returns a delayed function. The shifting property holds under the precise condition that f be a tempered distribution (see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution