Consider interval $[0,1]$ with its Borel $\sigma$-algebra and Lebesgue measure on it. It is known that $f$ is an integrable function on $[0,1]$. $\mathcal{F_n}=\sigma([\frac{k-1}{2^n},\frac{k}{2^n}))$ where $1\leq k \leq 2^n$. For $n=1,2,...$, $x\in [\frac{k-1}{2^n},\frac{k}{2^n})$ define
$$\mathbb{E}(X|\mathcal{F_n})(x):=\int_{\left[\frac{k-1}{2^n},\frac{k}{2^n}\right)} 2^n f \text{ d}\lambda$$
Then, what is $X$ ?
$\textbf{My try so far:}$
Straighforward for the definition of conditional expectation $\mathbb{E}(X|\mathcal{F_n})$, i found that $X=2^n f$. Yet, i then got confused, what is $n$ ? Shouldn't it be fixed $n$, right? Then, what is it?
Or am i missed something?