Let $$ X := \{ x = (x_1, \dots, x_n) \in \mathbb{R}^n \mid x_k \in \{0,1\} \forall k \in \{1,\dots,n\}\} $$ and let for all $k \in \{1, \dots, n\}$ $$ f_k : X \to \mathbb{R} \qquad \textrm{s.t.} \qquad f_k(x) = x_k $$ For each $j \in \{1, \dots, n\}$ describe explicitly the smallest $\sigma$-algebra $\mathcal{A}_j \in \mathcal{P}(X)$ such that all functionss $f_1, \dots, f_j$ are $\mathcal{A}_j$-$\mathcal{B}(\mathbb{R})$-measurable, i.e.: $$ \mathcal{A}_j := \bigcap_{\textrm{$\mathcal{A}$ is $\sigma$-alg. s.t. $f_1, \dots, f_j$ are $\mathcal{A}$-$\mathcal{B}(\mathbb{R})$-meas.}} \mathcal{A} $$
Show that $f_l$ is not $\mathcal{A}_j$-$\mathcal{B}(\mathbb{R})$-measurable when $l > j$.
Hi, whilst revising for an exam, I have attempted this with no luck. I thought about considering the $n$ dimensional lattice and also a formula for the number of sets in the $\sigma$-algebra. Help would be much appreciated.
Thanks.
Hint:
In general if $Y$ is a function $\Omega\to\mathbb R$ such that $\{Y\in C\}=\Omega$ for some countable set $C$ then: $$\sigma(X):=\{\{X\in D\}\mid D\subseteq C\}$$
Consequently if $C$ is finite with $\{Y=c\}\neq\varnothing$ for every $c\in C$ then the cardinality of $\sigma(X)$ is $2^{|C|}$.