then Lebesgue measure on formula_54 is equal to its own dual measure. This convention minimizes the number of factors of formula_66 that show up in various places when computing Fourier transforms or inverse Fourier transforms on Euclidean space. (In effect it limits the formula_66 only to the exponent rather than as a pre-factor outside the integral sign.) Note that the choice of how to identify formula_54 with its dual group affects the meaning of the term "self-dual function", which is a function on formula_54 equal to its own Fourier transform: using the classical pairing formula_70 the function formula_71 is self-dual. But using the pairing, which keeps the pre-factor as unity, formula_72 makes formula_73 self-dual instead. This second definition for the Fourier transform has the advantage that it maps the multiplicative identity to the convolution identity, which is useful as formula_30 is a convolution algebra. See the next section on the group algebra. In addition, this form is also necessarily isometric on formula_75 spaces. See below at Plancherel and "L"2 Fourier inversion theorems