limit of characteristic function and relation of primitive

Question

1. If I have $h(x)=1_A(x) \, f(x) = \begin{cases}f(x) & \text{if } x \in A \\ 0 & \text{if } x \not\in A\end{cases}$, where $A$ is an arbitrary set, then why it's equal to $f(x)$ almost everywhere?

2. Is there equality between an infinity norm of a function and its primitive?

Answer 1

1. It isn't in general. For example, with $f(x)\equiv 1$ and $A=\emptyset$ we have $1_A(x) \, f(x) \equiv 0 \neq f(x) \equiv 1$ everywhere.

2. No, there isn't. For example, $f_n(x) = \sin nx$ where $n=1,2,3,\ldots$ all have $\|f_n\| = 1$ but $\|f_n'\|=n$. Thus there is no such equality.