# Thread: proving a set is measurable

1. ## proving a set is measurable

Question:
Suppose that A is a subset of \mathbb{R} with the property that for every
\epsilon > 0 there are measurable sets B and C such that B \subset A
\subset C and m(C \cap B^c) < \epsilon. Prove that A is measurable.

My attempt:
Im finding this stuff really tricky! I know the definition of a set
being measurable but having real difficulty showing it. I know that i have
to use the fact that B and C are both measurable, but dont know what set to
use in the definition! help please! thanks

2. Originally Posted by ramdayal9
Question:
Suppose that A is a subset of \mathbb{R} with the property that for every
\epsilon > 0 there are measurable sets B and C such that B \subset A
\subset C and m(C \cap B^c) < \epsilon. Prove that A is measurable.

My attempt:
Im finding this stuff really tricky! I know the definition of a set
being measurable but having real difficulty showing it. I know that i have
to use the fact that B and C are both measurable, but dont know what set to
use in the definition! help please! thanks

Ok, what is the definition of "measurable set" that you have?

Tonio

3. A set K is measurable if for all E \subset R, m(E) = m(K \cap E) + m(K^c \cap E)

4. Originally Posted by ramdayal9
A set K is measurable if for all E \subset R, m(E) = m(K \cap E) + m(K^c \cap E)

This must be the OP I pressume: $E \subset R, m(E) = m(K \cap E) + m(K^c \cap E)$

5. Originally Posted by ramdayal9
Question:
Suppose that A is a subset of $\mathbb{R}$ with the property that for every
$\epsilon > 0$ there are measurable sets B and C such that $B \subset A
\subset C \,\,and \,\,m(C \cap B^c) < \epsilon$
. Prove that A is measurable.

My attempt:
Im finding this stuff really tricky! I know the definition of a set
being measurable but having real difficulty showing it. I know that i have
to use the fact that B and C are both measurable, but dont know what set to
use in the definition! help please! thanks
.