Posted: November 14th, 2015

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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