$\sigma$-algebras and measurable functions I have this exercise on an assignment sheet and I don't really understand what is asked. The formulation is as follows:
Let $X$ be the set of sequences of $n$ zeros and ones ($x\in X$ has the form $x=(x_1,...,x_n)$ where $x_k\in\{0,1\}$) and let $f_k:X\rightarrow\mathbb{R}$ the function $f_k(x)=x_k$. For each $j=1,...,n$, describe explicitly, determining all its sets, the $\sigma$-algebra $A_j$ defined as the smallest $\sigma$-algebra such that all functions $f_k$, $k=1,...,j$ are $A_j$-measurable. 
I don't get what I have to do so now I just know that $f_k$ is measurable if and only if $f_k^{-1}[(a,\infty)]$ is measurable. But this set is $\emptyset$ for $a\geq 1$, is $X$ for $a<0$ and is what for $a\in[0,1)$? 
Any suggestion is appreciated! Thank you
 A: If you have a measurable space $(Y, \mathcal Y)$ and some set $X$, and a family of functions $f_i : X \to Y, i \in I$ for some index set $I$, then the $\sigma$-algebra $\sigma(f_i : i \in I)$ generated by these functions on $X$ is the $\sigma$-algebra generated by the inverse images of the measurable sets in $\mathcal Y$ (as by definition these should be contained in $\sigma(f_i : i \in I)$ to make the functions measurable w.r.t. to this $\sigma$-algebra).
In your case the set $X$ is finite, and I suppose $\mathbb R$ is equipped with the Borel $\sigma$-algebra.
Suppose $n = 1$ (the general case is up to you). Then we can write $X = \{0,1\}$ (by indentifying the length $1$ sequenes with the set $X$) and for $f : X \to \mathbb R$ suppose $f(0) = a, f(1) = b$ with $a < b$, then 
\begin{align*}
 f^{-1}((-\infty,a-1]) & = \emptyset \\
 f^{-1}((-\infty,a]) & = \{0\} \\
 f^{-1}((a, b]) & = \{1\} \\
 f^{-1}((-\infty, b]) & = \{0,1\}
\end{align*}
and so we see that all sets must be contained in the generated measure space (as the sets from which I took the inverse images are contained in the Borel $\sigma$-algebra of $\mathbb R$), i.e. the generated measure space equals the discrete $\sigma$-algebra. 
Something like that works for arbitrary finite $X$, and in particular for your set of $n$-length sequences of zeros and ones. Hope that was helpful, let me know if you have any further questions!
