Random approximations for a deterministic optimization problem occur in many situations. Unknown parameters or probability distributions in real-life decision problems are replaced with estimates; more and more solution algorithms use random steps. Moreover, many estimation procedures in statistics are random optimization problems, which can be supplemented with a deterministic limit problem. Confidence sets for solution sets and level sets of a deterministic decision problems can be derived on the base of suitable uniform concentration-of-measure results for sequences of random functions. In the present paper approximations with kernel density estimators are considered and concentration-of-measure results for these estimators are provided. The results are employed to derive confience sets for modes and high density regions of probability distributions. Furthermore, applications in stochastic programming are investigated. It is shown how the assertions can be utilized to derive uniform concentration-of-measure results for chance constraints and functions which are expectations.