Definition of measurable set in $\mathbb{R}^n$.

Question

I have read the following definitions for a measurable set $A$:

1. There exists a sequence of step functions (functions of the form $\sum_{i=1}^m a_i 1_{I_{i}}$, where $I_i$ is an interval in $\mathbb{R}^n$) such that it converges to $1_A$ a.e.

2. There exists a sequence of simple functions (functions of the form $\sum_{i=1}^m a_i 1_{A_{i}}$, where $A_i$ is a Borel set) such that it converges to $1_A$ a.e.

3. $A$ is a Borel set.

4. For all $B\subseteq\mathbb{R}^n$, $m(A\cap B)+m(A^c\cap B)=m(B)$, where $m$ is the outer Lebesgue measure.

Which ones are equivalent?

My thought: $$ (2 \Leftrightarrow 3)\Rightarrow 1, $$ $$ 1\not\Rightarrow 3.$$ I would like to know if $4\Leftrightarrow 1$.