Posted: November 14th, 2015

# Measure Theory

Measure Theory

1. In this problem, we are working on R, with the Borel s-algebra.
(a) Show that the function f(x) = bxc is measurable.
(bxc is the biggest integer n for which n 6 x.)
(b) Show that every decreasing function is measurable.
2. On R, let A be the s-algebra {Ø, R,(-8, 0],(0, 8)}. Give an example, with proof of:
(a) A non-constant function that is measurable.
(b) A function that is not measurable.
3. Show that if f is a measurable function, then so is f
2
.
Is the converse true? Give a proof or a counterexample.
4. This question is about Bernoulli space, where the s-algebra E is generated by the
events E1, E2, . . . .
(a) Let X(?) be the number of Tails in the first two tosses. Show that X is a random
variable (i.e., that X is measurable).
(b) Show that the formula X(?) = 2?1 – 3?3 defines a random variable (measurable
function).

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