Confidence sets for modes or level sets of densities are usually derived from the asymptotic distribution of a suitable statistic. Mostly one does not have further information about how close the asymptotic distribution comes to the true distribution for a fixed sample size n. In order to derive conservative cofindence sets for each sample size recently an approach was suggested that does not need full information about a distribution, but instead employs a quantified version of semi-convergence in probability of random sets. The application of this approach to modes or level sets of density functions requires uniform concentration-of-measure results for the density estimators. The aim of the present paper is to prove a result of that kind for the multivariate kernel density estimator. The inequality is also of own interest as it provides a conservative confidence band for the density function.