manner. Let η be an absolutely integrable function on R of total integral 1, and define In n dimensions, one uses instead the scaling Then a simple change of variables shows that ηε also has integral 1. One may show that () holds for all continuous compactly supported functions f, and so ηε converges weakly to δ in the sense of measures. The ηε constructed in this way are known as an approximation to the identity. This terminology is because the space L1(R) of absolutely integrable functions is closed under the operation of convolution of functions: whenever f and g are in L1(R). However, there is no identity in L1(R) for the convolution product: no element h such that for all f. Nevertheless, the sequence ηε does approximate such an identity in the sense that This limit holds in the sense of mean convergence (convergence in L1). Further conditions on the ηε, for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence almost everywhere. If the initial is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing η to be a suitably normalized bump function, for instance In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η1 to be a hat function. With this choice of η1, one has which are all continuous and compactly supported, although not smooth and so not a mollifier. Probabilistic considerations In the context of probability theory, it is natural to impose the additional condition that the initial η1 in an approximation to the identity should be positive, as such a function then represents a probability distribution. Convolution with a probability distribution is sometimes favorable because it does not result in overshoot or undershoot, as the output is a convex combination of the input values, and thus falls between the maximum and minimum of the input function. Taking η1 to be any probability distribution at all, and letting as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments. For instance, if η1 is the uniform distribution on , also known as the rectangular function, then: Another example is with the