  A result of Chernoff gives sufficient condition for an $L^2$-function on
$\R^n$ to be quasi-analytic. This is a generalization of the classical
Denjoy-Carleman theorem on $\R$ and of the subsequent work on $\R^n$ by Bochner
and Taylor. In this note we endeavour to obtain an exact analogue of the result
of Chernoff for $L^p, p\in [1,2]$ functions on the Riemannian symmetric spaces
of noncompact type. No restriction on the rank of the symmetric spaces and no
condition on the symmetry of the functions is assumed.
