The Cartan Lemma by B.Ya. Levin and its Applications - Gorin, E. A. (Журнал математичної фізики, аналізу, геометрії, 2007)
Let X={X,d} be a complete separable metric space and let μ be a nonnegative regular borel measure on X such that μ(X)<∞. Let φ=φ(t) be a strictly increasing to infinity continuous real function on the semiaxis [0,∞), for which φ(0)=0. Assume B(a,t)def = {xX | d(x,a)<t}. There exist the results called a Cartan lemma and in which is estimated a massiveness of such sets xX where the conditionμ(B(x,t))<φ(t) for all t>0, is not fulfilled.We give one of the delivered appeared while studing of Levin’s lectures in Moscow in 1970. Also we give applications to some problems of the finite-dimensional and infinite-dimensional analysis. The peculiarity of our approach is: such parameters as that dimension do not appear in the initial estimates (at least explicitly).Generally, the article is a review, however some of the given results have not been published yet. Mathematics Subject Classification: 30-02, 31-02.