I have a trouble with Exercise 1.9 of by Lieb and Loss.
Let $(\Omega, \Sigma, \mu)$ be a measure space. In this book, a nonnegative $\mu$-measurable function $f$ is integrable if the (improper) Riemann integral $ \int_{0}^{\infty} F_{f}(t) dt$ is finite where $F_{f}(t) = \mu \{ x \in \Omega : f(x) > t \}$.
Exercise 1.9 is about proving the linearity of Lebesgue integral.
In the part (b), I have to construct functions that satisfy:
Suppose $f$ is a nonnegative integrable function. For any integer $N$, find functions $f_{N}$ that take only finitely many values, such that $| \int f - \int f_{N} | \le C/N$ for some constant $C$ independent of $N$.
I tried the usual simple function that approximates f: $$ f_{N} = \sum_{k=1}^{N 2^{N}} \frac{k-1}{2^{N}} \chi_{\frac{k-1}{2^N} < f \le \frac{k}{2^{N}}} + N \chi_{f>N}$$
Here $\chi$ denotes the characteristic function.
However, I was only able to prove that $f_{N} \uparrow f$ and $\int f_{N} \uparrow \int f$. So I am able to prove the linearity of Lebesgue integral, but I have no idea about how to construct $f_{N}$ such that $\int f$ and $\int f_{N}$ only differs in order $N^{-1}$.
Any help is appreciated. Thank you.