Is it possible to construct an uncountable partition $(A_\delta)_{\delta\in[0,1]}$ of the unit interval $[0,1]$ with the following properties:

1) $\cup_{\delta\in[0,1]} A_\delta=[0,1]$;

2) If $\delta_1\neq \delta_2$, then $A_{\delta_1}\cap A_{\delta_2}=\emptyset$;

3) For each $\delta$ we have $\mu (A_\delta)=1$, where $\mu$ stands for the outer Lebesgue measure.

My motivation is as follows. Using Vitali sets of outer measure $1$, it is possible to construct a **countable** partition of $[0,1]$ with such properties. Indeed, one just need to put 
$$A_\delta:=V+\delta \mod 1,\quad \delta\in\mathbb{Q},$$
where $V$ is the Vitali set of outer measure $1$.  

My question is whether it is possible to construct an **uncountable** partition of $[0,1]$ with these properties.

Thanks!