Some properties of lacunary series and Gaussian measures connected with uniform versions of properties of Egoroff and Lusin

B. S. Tsirel'son

Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 3, pp. 664-667

Résumé

Let $A$ be a measurable subset of $[0,1]$ and $\operatorname{mes}A>0$. For any function $f$ satisfying \begin{gather*} f(t)=\sum(a_k\cos\lambda_kt+b_k\sin\lambda_kt),\quad\lambda_1,\lambda_2,\dots>0,\quad\inf(\lambda_{k+1}/\lambda_k)>1, \\ \sum(a_k^2+b_k^2)<\infty\quad\text{and }|f(t)|\le1\quad\text{a.e.\ on }A, \end{gather*} we can find a sequence of sets $B_1\subset B_2\subset\dots\subset[0,1]$, $\operatorname{mes}B_n\to1$, and a function $F\in L_1[0,1]$ such that $\sum(a_k\cos\lambda_kt+b_k\sin\lambda_kt)$ converges uniformly on every $B_n$ and $|f(t)|\le F(t)$ a.e. on $[0,1]$. The sequence $\{B_n\}$ and the function $F$ depends on $\{\lambda_k\}$, $A$ only. The function $F$ may be chosen in such a way that $\int_0^1\exp(\alpha F(t))\,dt<+\infty$ for some positive $\alpha$. It is interesting to observe an analogy between this theorem and similar results about Gaussian random variables.