Real analysis-Quiz II

 

Let  A={x,y,z,w} and B={1,2,3} . Which of the following sets are functions with domain A and codomain B?

• {(x, 1), (y, 2), (z, 3)}

• {(x, 1), (y, 2), (z, 3), (w, 1)}

• {(x, 1), (x, 2), (y, 1), (z, 1), (w, 1)}

The set of integers Z is

• closed

• perfect

• open

• bounded

  Let f: R→R and let

 

 1) {x∊ R: f(x)> a} is measurable for all a∊ R

2) {x∊ R: f(x)= a} is measurable for all a∊ R

Then

•    (1) implies (2)

•    (2) implies (1)

• (1) and (2) are equivalent

 

Let the nonnegative function f be integrable over E. Then

• f is continuous  a.e. on E

• f is finite a.e. on E

•   f is equal 0 a.e. on E

 

  There exists a set of positive measure whose every subset is measurable.

 

• True

• False

   Outer measure is finite additive.

• True

• False

If a function g: R→R is measurable, then the set

 

{x∊ R: g(x) < b} is measurable for all b∊ R.

• False

• True

The intersection of two measurable sets is

• measurable

• countable

• non-measurable

• finite

Let C be a countable collection of measurable sets. Then the union

 

of all sets from C is

• measurable

• finite

• bounded

• non-measurable

A singleton set {a}  is a measurable set with a positive measure.

• True

• False