On the theorem of Frullani

Abstract

We prove that, for every function f : R + → C f:{{\mathbf {R}}^ + } \to {\text {C}} such that ( f ( a x ) − f ( b x ) ) / x (f(ax) - f(bx))/x is Denjoy-Perron integrable on [ 0 , + ∞ ) [0, + \infty ) for every pair of positive real numbers a , b a,b , there exists a constant A A (depending only on the values of f ( t ) f(t) in the neighborhood of 0 and + ∞ + \infty ) such that \[ ∫ 0 ∞ f ( a x ) − f ( b x ) x d x = A l o g a b . \int _0^\infty {\frac {{f(ax) - f(bx)}}{x}} dx = Alog \frac {a}{b}. \] To prove this assertion, we identify a Denjoy-Perron integrable function f : R → C 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 f is the Denjoy-Perron integral of f f . Assuming the Continuum Hypothesis, we construct an example of a non-Lebesgue measurable function that satisfies the hypotheses of the first theorem.