measure theory
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In this problem, we are working on R, with the Borel -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 -algebra f?;R; (-1; 0]; (0;1)g. 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 f2.
Is the converse true? Give a proof or a counterexample.
4. This question is about Bernoulli space, where the -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).
Deadline: To be handed in to the School office by 5 pm on Monday October 31st.
Groupwork: If you do this assignment with a friend, please give their name in the box
below. They must do the same on their form. You may be asked to explain your answers
in order to ensure that one of you did not simply copy the other’s work. You should write
up your solution in your own words and using your own notation.
Do not copy! Identical solutions are not acceptable and will be penalized
