Real Analysis (Elementary sets)

1). Let M be an elementary set. Prove that | closure(M)M | = 0. ( closure of M can also be written as M bar, and it is the union of M and limit points of M). 2). If M and N are elementary sets then show that | M union N | + | M intersection N| = |M| + |N| The def of elementary set : If M is a union of finite members of disjoint cells, then M is said to be an elementary set. I believe it is related to lebesgue measure topics, but not so sure.

Limit

Please be as explicit as possible with the solution steps. Thank you! --- Find the limit and justify your answer: (see attachment) ---

L^1

Please see attached

Borel measurable (Borel functions)

1).Let f(X) : R -> R be the following: f(x) = { 1 if x is in Q (rationals) , 0 if x is not in Q ( irrational)} Prove that f(x) is Borel measurable ( Borel functions).

Borel measurable (Borel functions)

Let f(x) be { 1/x if x is not 0. and 1 if x = 0} . Prove that f(x) is borel function ( borel measurable).