Radon's inversion formulas

Radon showed the pointwise validity of his celebrated inversion formulas for the Radon transform of a function $f$ of two real variables (formulas (III) and (III$'$) in J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. kl. 69 (1917), 262-277) under the assumption that $f$ is continuous and satisfies two other technical conditions. In this work, using the methods of modern analysis, we show that these technical conditions can be relaxed. For example, the assumption that $f$ be in $L^p(\mathbb{R} ^2)$for some $p$ satisfying $4/3<p<2$ suffices to guarantee the almost everywhere existence of its Radon transform and the almost everywhere validity of Radon's inversion formulas.