Partition of real numbers into dense subsets of positive measure [duplicate]

Question

Does there exist a partition of real numbers (with standard topology; Lebesgue measure) into two measurable sets $A$ and $B$, satisfying the following properties:

1. $A$, $B$ are both dense in the real numbers.
2. $A$, $B$ both have positive measure.

Edited: as pointed out by Henry, actually I am looking for A B such that for any open interval I, the intersection of I and A, the intersection of B and I both have positive measure. Sorry for the confusion.

Answer 1

A=Positive rational numbers and negative irrational numbers.

B=Negative rational numbers, positive irrational numbers and 0.

Answer 2

There is a rather famous construction of such a set. It helps to know the following facts:

1. if $O$ is a nonempty open set then there exists a nowhere dense closed set $C \subset O$ with the property that $0 < m(C)$, and
2. if $C_1,\ldots,C_n$ is a collection of closed nowhere dense sets and $O$ is a nonempty open set, then $O \setminus (C_1 \cup \cdots \cup C_n)$ is nonempty and open.

In particular, if $O$ is open we can find two disjoint nowhere dense closed sets $C_1,C_2 \subset O$ with the property that $0 < m(C_1)$ and $0 < m(C_2)$: just select $C_2 \subset O \setminus C_1$.

Let $\{I_n\}$ be a sequence of open intervals that forms a basis for the topology of the line.

Step 1: Select two disjoint nowhere dense closed sets $K_1,K_2 \subset I_1$ such that $0 < m(K_1)$ and $0 < m(K_2)$.

Step 2: Select two disjoint nowhere dense closed sets $K_3,K_4 \subset I_2 \setminus (K_1 \cup K_2)$ such that $0 < m(K_3)$ and $0 < m(K_4)$.

Step 3: Select two disjoint nowhere dense closed sets $K_5,K_6 \subset I_3 \setminus (K_1 \cup K_2 \cup K_3 \cup K_4)$ such that $0 < m(K_5)$ and $0 < m(K_6)$.

Step 4: Proceed inductively to obtain a sequence $\{K_j\}$ of pairwise disjoint nowhere dense closed sets with the property that $K_{2j-1},K_{2j} \subset I_j$. Define $$E = \bigcup_j K_{2j-1},\quad F = \bigcup_j K_{2j}.$$

That ends the construction. If $O \subset \mathbb R$ is open, there exists an interval $I_j \subset O$ so that $$ E \cap O \supset E \cap I_j \supset K_{2j-1}$$ and $$ E^c \cap O \supset F \cap O \supset F \cap I_{j} \supset K_{2j}.$$ Thus $$m(E \cap O) \ge m(K_{2j-1}) > 0 \quad \text{and} \quad m(E^c \cap O) \ge m(K_{2j}) > 0.$$

We conclude that both $E$ and $E^c$ occupy a set of positive measure in every open subset of the line.