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 $\mu$ be a nonnegative regular borel measure on $X$ such that $\mu(X)<\infty$. Let $\varphi=\varphi(t)$ be a strictly increasing to infinity continuous real function on the semiaxis $[0,\infty)$, for which $\varphi(0)=0$. Assume $B(a,t)\stackrel{\mathrm{def}}{=} \{x\in X\; |$ $ d(x,a)<t\}$. There exist the results called a Cartan lemma and in which is estimated a massiveness of such sets $x\in X$ where the condition $$\mu(B(x,t))<\varphi(t) \;\mathrm{ for \; all }\; t>0,$$ 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.