Here is the step that I do not understand:
Should not the equality before the last contains a power of $p$ over the sum and an integral sign according to the definition of p-norm , or maybe I do not understand how it comes from its preceding one, could anyone help me understand this please?
If $F$ is a measurable set, then $\|\chi_F\|_p^p = \int \chi_F^p \, d\mu = \int \chi_F \, d\mu = \mu(F)$ because $\chi_F^p = \chi_F$.
The context of your picture is missing, but if I presume that $E_1,E_2,\ldots$ are disjoint sets whose union is $E$, then $\chi_E - \sum_{k=1}^n \chi_{E_k} = \chi_{\bigcup_{k=n+1}^\infty E_k}$, from which you obtain the desired equality.