I'm studying the book Higher Order Fourier Analysis by Terence Tao (https://terrytao.files.wordpress.com/2011/03/higher-book.pdf). There, it defines that a function $f:[N]\to\mathbb{C}$ has Fourier complexity at most M if it can be expressed as $$f(n) = \sum_{m=1}^{M'}c_me(\alpha_mn)$$ for some $M\leq M$ and $c_1,\ldots,c_{M'}\in\mathbb C$ of magnitude at most 1.

It notes that from the Fourier inversion formula, every function has finite Fourier complexity, but it may grow with N.

But then, it says that, in order to work with indicator functions we need to take the $L^1$ closure and work with the Fourier mesurable functions.

Then, it defines that a function $f:[N]\to \mathbb C$ is Fourier mesurable with growth function $\mathcal F:\mathbb R^+ \to \mathbb R^+$ if, for every $K>1$ one can find a function $f_K:[N]\to\mathbb C$ of fourier complexity at most $\mathcal F(K)$ such that $E_{n\in[N]}|f(n)-f_K(n)|\leq 1/K$.

I don't understand why we need this concept (witch doesn't depend on $N$) if every function has finite Fourier complexity and we don't need to approximate it by this kind of functions. What do I not understand here?