Zero-one laws for polynomials in Gaussian random variables: A simple proof (Q1356620)

Let \(E\) and \(F\) be separable Fréchet spaces, let \(P(x)= s_0+ \sum^d_{k=1} S_k(x,\dots,x)\) be a polynomial of order \(d\), where \(s_0\in F\) and \(S_k\) is a symmetric measurable form on \(E^k\) with values in \(F\), \(1\leq k\leq d\), and let \(\mu\) denote a Gaussian centered measure on \(E\). The main result states that if \(V\) is a linear measurable subspace of \(F\), then \(\mu\{P\in V\}= 0\) or 1. In a natural way, this result is applied to Gaussian chaoses of finite order.