For a vector space $X$ and $d\in\mathbb N$ a function $f:X \to
{\mathbb {R}}$ is called a polynomial on $X$ of degree $d$ if there
exist multilinear functions $f_k:X^k\to {\mathbb {R}}$, $k=1,
\ldots, d$, and $f_0 \in \mathbb{R}$ such that
\[
f(x) = f_d(x,\ldots,x) + f_{d-1}(x,\ldots,x) + \cdots + f_1(x) + f_0
\] for $x\in X$.
In my talk I will address the following questions.
Let $(X, \mathcal B, \mu)$ be a probability space, where
$X$ is a locally convex space, $\mathcal B$ is the Borel $\sigma-$field and $\mu$ is a Radon Gaussian
measure on $\mathcal B$. Consider a sequence $(f_k)_{k\in\mathbb {N}}$ of
continuous polynomials on $X$ of degree $d$, which converges to a
function $f$ in measure $\mu$. Does there exist a polynomial $f^*$
on $X$ of degree $d$ such that $f = f^*$ almost everywhere?
Next, let $n\in\mathbb N$ and consider the probability space
$(X^n,\mathcal B^{\otimes n},\mu^{\otimes n})$. Let
$(g_k)_{k\in\mathbb {N}}$ be a sequence of continuous multilinear
functions on $X^n$, which converges to a function $g$ in measure
$\mu^{\otimes n}$. Does there exist a multilinear function $g^*$ on
$X^n$ such that $g = g^*$ almost everywhere?
It turns out, that the answers to both questions are positive, which
follows from much more general assertions.
The talk is based on joint work with Lavrentin
Arutyunyan, Lomonosov Moscow State University.