Universal characteristic factors and Furstenberg averages

Let $X=(X^0,\mathcal{B},\mu,T)$ be an ergodic probability measure-preserving system. For a natural number $k$ we consider the averages

$\tag{*} \frac{1}{N}\sum_{n=1}^N \prod_{j=1}^k f_j(T^{a_jn}x)$

where $f_j \in L^{\infty}(\mu)$, and $a_j$ are integers. A factor of $X$ is characteristic for averaging schemes of length $k$ (or $k$-characteristic) if for any nonzero distinct integers $a_1,\ldots,a_k$, the limiting $L^2(\mu)$ behavior of the averages in (*) is unaltered if we first project the functions $f_j$ onto the factor. A factor of $X$ is a $k$-universal characteristic factor ($k$-u.c.f.) if it is a $k$-characteristic factor, and a factor of any $k$-characteristic factor. We show that there exists a unique $k$-u.c.f., and it has the structure of a $(k-1)$-step nilsystem, more specifically an inverse limit of $(k-1)$-step nilflows. Using this we show that the averages in (*) converge in $L^2(\mu)$. This provides an alternative proof to the one given by Host and Kra.