I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The set of all functions $\varphi : \Lambda \to \mathbb{R}$ is isomorphic to $\mathbb{R}^{|\Lambda|}$, so that these functions are represented as vectors $\varphi = (\varphi\_{x})\_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$, and are called *fields*. Given $\xi\_{1},...,\xi\_{N} \in \mathbb{R}^{|\Lambda|}$, we set $\varphi\_{j} := \sum\_{k> j}\xi\_{k}$ for each $j=0,...,N-1$ and $\varphi\_{N} = 0$. 


Now the author states the following: "Given $X\subset \Lambda$, let $\mathcal{N}\_{j}(X)$ be the algebra of functions measurable with respect to the $\sigma$-algebra generated by $\{\varphi\_{j}(x), \hspace{0.1cm} x \in X\}$. In more down to earth terms, an element of $\mathcal{N}\_{j}(X)$ is a function only of fields at points $x \in X$."


I don't understand what is this $\sigma$-algebra. It seems confusing to me. What does it mean? 


**Edit:** The text can be found in 
math.ubc.ca/~db5d/Seminars/PCMILectures/lectures.pdf



Imprecise Definition of a $\sigma$-algebra
$\newcommand\vpi{\varphi}$
$\newcommand\La{\Lambda}$
$\newcommand\R{\mathbb R}$
You misunderstood the notes: For each $j\in\{0,\dots,N\}$ and each $x\in\La$, $\vpi\_j(x)$ is (not a real number but) a real-valued random variable (r.v.). 


Indeed, (i) formula (2.17) on page 25 in the notes implies that $\vpi\_j=\sum\_{k=j+1}^N\zeta\_j$ (with $\vpi\_N=0$), (ii) line 2 on page 22 there tells us that $\zeta\_j\sim N(C\_j)$, and (iii) by formula (1.20) on page 10, $\zeta\_j\sim N(C\_j)$ means that $\zeta\_j=(\zeta\_j(x)\colon x\in\La)$ is a random zero-mean Gaussian vector (with values in $\R^\La$ and) with the covariance matrix $C\_j=(C\_j(x,y))\_{(x,y)\in\La^2}$. 


So, for each $j\in\{0,\dots,N\}$ and each $X\subseteq\La$, the $\sigma$-algebra in question is the $\sigma$-algebra generated by the set $\{\varphi\_j(x)\colon x\in X\}$ of real-valued r.v.'s, that is, the smallest $\sigma$-algebra with respect to which all the r.v.'s $\varphi\_j(x)$ with $x\in X$ are measurable. 


