Question

SHow that a union of a finite or countable number of sets of lebesgue measure zero is a set of lebesgue measure zero.

Please show all steps

Answer #1

A countable union of disjoint countable sets is countable
Note: countable sets can be either finite or infinite
A countable union of countable sets is countable
Note: countable sets can be either finite or infinite

Prove the union of a finite collection of countable
sets is countable.

Use the fact that “countable union of disjoint countable sets is
countable" to prove “the set of all polynomials with rational
coefficients must be countable.”

Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai
countable, then union i∈I Ai is countable. Hints: (i) Show that it
suffices to prove this for the case in which I = N and, for every i
∈ N, the set Ai is nonempty. (ii) In the case above, a result
proven in class shows that for each i ∈ N there is a...

prove that the lebesgue measure on R has a countable
basis

Prove that tue union of countable sets is countable.

Prove the union of two infinite countable sets is countable.

Show that the following property holds for countable sets:
if S_1, S_2,S_3,... is a sequence of countable sets of real
numbers, then the set S formed by taking all the elements that
belong to at least one of the sets S_i is also a countable set.

Prove that the union of infinitely many sets is countable using
induction.

Prove that the set of all finite subsets of Q is countable

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