Several results on smoothness of functions of the type \(t\to \mu (x\in X:\) \(F(x)<t)\), where \(\mu\) is a measure in infinitely dimensional space X, are presented without proofs. The cases when F is a polynomial or F is a functional of integral type with analytic kernel on the path space of a random process are dealt with. In particular, the T. S. Pitcher’s conjecture [Trans. Am. Math. Soc. 101, 168-176 (1961; Zbl 0121.132)] is proved.