[Math4u] Lebesgue density

Define Lebesgue density of a set E (a subset of the real numbers R)
at x as den(x) = limit m(E intersect [x-h, x+h])/(2h) as h ->0 where m
(A) is the Lebesgue measure of a set A and [a, b] is the usual
notation for closed interval from a to b. For example, if E = [0, 1]
(or (0,1) or [0,1) or (0,1]), then den(0) = 1/2 = den(1), den(x) = 1
if 0<x<1 and den(x) = 0 if x < 0 or x > 1.

The problem is given any t with 0<t<1, does it exist a set E such
that den(0) = t. I think it does. But I need somebody to verify it:

My goal is just to construct a set E such that den(0) = t for some t
between 0 and 1 but not = to 1/2. If I succeed at that, I can modify
E so that it works for any t between 0 and 1.

My set is E = Union [2/3^(n+1), 1/3^n] as n = 0 to infinity. For this
E, clearly m(E) = 1/2. Therefore, as h -> 0,
lim m(E intersect [-h, h])/(2h)
= lim m(E intersect [0, h])/(2h)
= lim ((1/2)*h)/(2h)
= 1/4

My problem is I can't convince myself that m(E intersect [0, h]) is
(1/2)*h, however I think it is when h -> 0. I just need somebody to
show me my above limit is correct with a stronger reason.

Thank you.

Hai He

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