(c) Show that any countable subset of R is measurable and of Lebesgue measure zero. (d) Use the previous items to justify that if a < b then the interval (a, b) is not countable. (e) Suppose that E is measurable, E CB CR and m* (B) = m* (E) < x. Prove that B is measurable.
Sol:-
(c) Any countable subset of R can be written as {a1, a2, a3, ...}, where ai ∈ R for all i. To show that this set is measurable and has Lebesgue measure zero, we can define a sequence of intervals In = (an - 1/2n, an + 1/2n) for n ∈ N. Then the countable subset can be written as the union of these intervals:
{a1, a2, a3, ...} = ⋃ n∈ N In
Since the Lebesgue measure of each interval is 1/2{n-1}, the Lebesgue measure of the countable subset is at most the sum of the measures of the intervals:
m({a1, a2, a3, ...}) ≤ ∑n ∈ N m(In) = ∑ n ∈ N 1/2{n-1} = 1
Therefore, the countable subset has Lebesgue measure zero. Since the countable subset is a countable union of intervals, which are measurable sets, it follows that the countable subset is also measurable.
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