I have a question about the following equality:
The following paragraph is clear:
Let $\Omega \subset \mathbb{R}^n$ be an open subset and $A \subset \Omega$ a subset with finite measure. Then there exists a set B, which is a finite, disjoint union of boxes, such that, for $\epsilon>0$ and $\lambda$ being the Lebesgue measure it follows, that $\lambda(A\Delta B)<\epsilon$.
Next, let $1\leq p < \infty$,$~$ $A,B\subset \Omega$ be as above and consider the characteristic functions $\chi_A$ and $\chi_B$. Then the Lebesgue norm $||\chi_A - \chi_B||_p^p=\lambda(A\Delta B)<\epsilon^p$.
How does the last equality follow?
We have $$ \Vert \chi_A - \chi_B \Vert_p^p = \Vert \chi_{A \Delta B} \Vert_p^p = \int \vert \chi_{A\Delta B} \vert^p \ d\lambda = \int \vert \chi_{A\Delta B}\vert \ d\lambda = \lambda(A\Delta B) < \varepsilon. $$ The additional power $p$ in what you wrote seems to be wrong.
Added: Using the definition of characteristic function we get $$ \vert \chi_A(x) - \chi_B(x)\vert = \begin{cases} \vert 0 - 0\vert,& x\in (A\cup B)^c,\\ \vert 1-1 \vert ,& x\in A \cap B,\\ \vert 1-0 \vert ,& x\in A\setminus B, \\ \vert 0 - 1 \vert,& x\in B \setminus A, \end{cases} = \begin{cases} 1,& x\in (A\setminus B)\cup (B\setminus A)\\ 0,& x\notin (A\setminus B)\cup (B\setminus A). \end{cases} $$ However, we have $(A\setminus B)\cup (B\setminus A)= A\Delta B$ and thus we get $$ \vert \chi_A(x) - \chi_B(x)\vert = \vert \chi_{A\Delta B}(x) \vert. $$