Prove (a), (b) a)If f is measurable, then f 2 measurable. (b) If{fn} is a sequen

Question

Prove (a), (b)
a)If f is measurable, then f2 measurable.

(b) If{fn} is a sequence of measurable functions, g(x) = inf fn(x); and
h(x) = lim inf fn(x), then g and h are measurable.

YOU MUST USE THE THEOREM AND DEFINITION TO PROVE THE TWO PARTS.
DEFINION:

Let f be a function defined on the measurable space X, with values in the
extended real number system. The function f is said to be measurable if
the set
{ x I f (x) > a }
is measurable for every a In R.
and by using the THEOREM

Each of the following four conditions implies the other three:

.

is measurable for every real a.

is measurable for every real a.

is measurable for every real

is measurable for every real a.

Explanation / Answer

Let n and r be non-negative integers with r<=n. Then prove that C(n,r)=C(n,n-r).

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