We consider a random measure $\mu $ on the spere S. One calls the measure $\mu $ invariant if ${\bf M}\mu (A)\mu (B) = {\bf M}\mu (gA)\overline {\mu (gB)} $ for every rotation g of the sphere.The following formula is proved: \[ \mu (A) = \sum\limits_{m = 0}^\infty {\sum\limits_{l = - m}^m {\xi _{m^l } \int_S {\overline {Y_m^l (x)} I_A (x)dx,} } } \] where ${\xi _m^l }$ are random variables, ${\bf M}\xi _m^l \xi _{m_1 }^{l_1 } = 0$ if $l \ne l_1 $ or $m \ne m_1 $, \[ {\bf M}\left| {\xi _m^{ - m} } \right|^2 = {\bf M}\left| {\xi _m^{ - m + 1} } \right|^2 \, = \cdots = {\bf M}\left| {\xi _m^m } \right|^2 = a_m ,\quad 0 \leqq a_m \leqq c <\infty ; \]$Y_m ^l (x)$are the spherical functions \[ I_A (x) = \left\{ {\begin{array}{*{20}c} {0,x \notin A,} \\ {1,x \in A;} \\ \end{array} } \right. \]x is a point of the sphere.