a simple function is approximated by a step function with respect to $L^p$. (proof verification)

Question

This proposition tells that there exists a step function that can approximate a simple function. But, I don't understand one step in this proof: $||\chi_A - \chi_{\mathcal{U}}||_p = [m(A \Delta \mathcal{U})]^{1/p}$.

I think $||\chi_A - \chi_{\mathcal{U}}||_p = (\int |\chi_A - \chi_{\mathcal{U}}|^p)^{1/p}.$ But, how can $\int |\chi_A - \chi_{\mathcal{U}}|^p = m(A \Delta \mathcal{U})$?

Answer 1

Think about what $\chi_A - \chi_U$ is. We have $$\chi_A(x) - \chi_U(x) = \begin{cases} 0 & x \in A \cap U \text{ or } x \notin A \cup U \\ 1 & x \in A \setminus U \\ -1 & x \in U \setminus A \end{cases}$$ So $\int |\chi_A - \chi_U|^p = \int_{A \setminus U} 1 + \int_{U \setminus A} 1 = m(A \setminus U) + m(U \setminus A) = m([A \setminus U] \cup [U \setminus A]) = m(A \Delta U)$.