Is there two sets A, B such that both are of measure zero/null, and A+B = [0,1]

A + B = {z : z = x + y, x in A, y in B}

(Note: a set of measure 0 has no interior points.)

If C is the Cantor "middle thirds" set, a closed subset of [0,1] of measure 0,
then C + C =[0,2]


Fact: x is in C <==> the base 3 representation of x has no 2's.
For example 1/3 = (0.1)_base 3 is in C. 1/3 + 1/3 = 0.2 is in C+C.

Likewise 0.02, 0.002, 0.22210122 = 0.11110111 + 0.11100011 all belong to C+C

Moreover C+C is the continuous image of the compact set C x C via the
function f(x,y) = x + y; so C+C is a compact, so closed, subset of [0,2].
1 is in C so 2 is in C+C. Every number in [0,2] is a limit of numbers
with finite ternary expansions 0.a_1 a_2 ... a_n or 1.b_1....b_m

The latter numbers belong to the closed set C + C and are dense in [0,2].
Therefore C+C = [0,2].