As an example, suppose formula_52, so we can think about formula_3 as formula_54 by the pairing formula_55 If formula_27 is the Lebesgue measure on Euclidean space, we obtain the ordinary Fourier transform on formula_54 and the dual measure needed for the Fourier inversion formula is formula_58. If we want to get a Fourier inversion formula with the same measure on both sides (that is, since we can think about formula_54 as its own dual space we can ask for formula_50 to equal formula_27) then we need to use