The discrete Fourier series in Peskin and Schroeder (page 285) I'm working on the discrete Fourier series in Peskin (page 285)，but I have two questions.

Question 1:
I tried to derive Eq.(9.21):
Consider $$
f(x)=\int \frac{d^{4} k}{(2 \pi)^{4}} e^{-i k \cdot x} \tilde{f}(k)
$$
So we have \begin{align}
\phi(x_i)&=\int \frac{d^{4} k}{(2 \pi)^{4}} e^{-i k \cdot x_i} \phi(k)\\&=
\lim _{n \rightarrow \infty}
\sum_{n} \frac{\Delta^{4} k_n}{(2 \pi)^{4}} e^{-i k_n \cdot x_i} \phi(k_n)
\end{align}
In finite volume we have $k_n=\frac{2\pi n}{L}$,so $$\Delta^{4} k_n=\frac{(2\pi)^4}{L^4}=\frac{(2\pi)^4}{V}$$
So the Fourier transform of finite volume is
\begin{align}
\phi(x_i)&=\sum_{n} \frac{\Delta^{4} k_n}{(2 \pi)^{4}} e^{-i k_n \cdot x_i} \phi(k_n)\\&=
\sum_{n} \frac{(2\pi)^4}{(2 \pi)^{4}V} e^{-i k_n \cdot x_i} \phi(k_n)\\&=\frac{1}{V}
\sum_{n} e^{-i k_n \cdot x_i} \phi(k_n)
\end{align}
This is consistent with Eq.(9.21). However, I didnʻt use $\left|k^{\mu}\right|<\pi / \epsilon$. Where does it come from？Is my derivation wrong？
Question 2:
I tried to derive this equation but I have difficulties $$
\mathcal{D} \phi(x)=\prod_{k_{n}^{0}>0} d \operatorname{Re} \phi\left(k_{n}\right) d \operatorname{Im} \phi\left(k_{n}\right).
$$
Consider Eq.(9.20)
\begin{align}
\mathcal{D} \phi&=\prod_{i} d \phi\left(x_{i}\right)\\&=\prod_{i} d 
\left(\frac{1}{V} \sum_{n} e^{-i k_{n} \cdot x_{i}} \phi\left(k_{n}\right)\right)
\end{align}
I don't know how to go on. Now how do I derive this equation？
 A: Lets start with your first question. Peskin and Schroeder are putting all of the fields in a box with volume $V = L^4$ in Euclidian Space (after Wick Rotation). This means that smallest Fourier mode the fields can have is $k^{\mu} = \frac{2\pi}{L}$. However, that box is discretized into a lattice with lattice spacing $\epsilon$. This lattice spacing brings with it a UV cutoff to all of the fields, coarse graining the fields so that the largest Fourier mode the fields can have is $k^{\mu} = \frac{\pi}{\epsilon}$. Having both a lattice spacing and a finite box means your theory have UV and IR cut offs. So the Fourier Integrals you have should really be written as
$$
\phi(x_i) = 
\int_{-\pi/\epsilon}^{-\pi/L}\frac{d^4k}{(2\pi)^4}e^{-ik_ix^i}\tilde{\phi}(k_i) 
+
\int^{\pi/\epsilon}_{\pi/L}\frac{d^4k}{(2\pi)^4}e^{-ik_ix^i}\tilde{\phi}(k_i)
$$
Usually you take $L\rightarrow \infty$, but that is where the $\pi/\epsilon$ comes from. 
The answer to your second question I believe is found here Does "sum over all paths" in the path integral imply "sum over all paths" in momentum space when one Fourier-transforms?.
