On the theorem of Frullani

Abstract:

We prove that, for every function $f:{{\mathbf {R}}^ + } \to {\text {C}}$ such that $(f(ax) - f(bx))/x$ is Denjoy-Perron integrable on $[0, + \infty )$ for every pair of positive real numbers $a,b$, there exists a constant $A$ (depending only on the values of $f(t)$ in the neighborhood of 0 and $+ \infty$) such that \[ \int _0^\infty {\frac {{f(ax) - f(bx)}}{x}} dx = Alog \frac {a}{b}.\] To prove this assertion, we identify a Denjoy-Perron integrable function $f:{\mathbf {R}} \to {\text {C}}$ with a distribution. In this way, we obtain the main result of this paper: The value at 0 (in Lojasiewicz sense) of the Fourier transform of the distribution $f$ is the Denjoy-Perron integral of $f$. Assuming the Continuum Hypothesis, we construct an example of a non-Lebesgue measurable function that satisfies the hypotheses of the first theorem.