A correction theorem for functions with integral smoothness

A theorem similar to the correction theorem of K. Oskolkov is proved. Namely, for a function with a given $k$th modulus of continuity calculated in a symmetric space $X$, for every $\epsilon>0$ a set is presented whose measure is at least $1-\epsilon$ and on which a sharp quantitative estimate of the uniform $k$th modulus of continuity of this function is given. It is shown that this estimate depends only on $\epsilon$ and on the fundamental function of the symmetric space.