Abstract: Harish-Chandra have got a Fourier inversion formula for C_c^∞(SL(2, R)). In this paper, we give a discussion on approximation identity kernels on SL(2, R) and get some properties of their Fourier transforms, and then, making use of these properties and Harish-Chandra's result, we prove that the Fourier inversion formula obtained by Harish-Chandra is also valid for C_b³ (SL'(2, R)).