Math Problems Vault

Problem 3.1.1

Write out the details of the proof, using diagonalization, that $\mathcal{B}$ is uncountable.

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Problem 3.1.2

Prove that $\mathcal{B}_{T}$ is countable, and use this result to give an alternative proof (as outlined in the text) that $\mathcal{B}$ is uncountable.

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Problem 3.1.3

(a)(b)

In this exercise you are asked to determine the set $B_{E}$ for given events $E$ .

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(a)

Determine the set $B_{E}$ if $E$ is the event that in the first three tosses, exactly two heads are seen. What is the probability of this event occurring? Does your answer to the last question agree with what you think the Lebesgue measure of $B_{E}$ should be?

(b)

Determine the set $B_{E}$ if $E$ is the event that in the first $n$ tosses, exactly $k$ heads are seen. What is the probability of this event occurring? Does your answer to the last question agree with what you think the Lebesgue measure of $B_{E}$ should be? Explain.

Problem 3.1.4

Prove that $(0,1] \backslash S$ is uncountable, where $S$ is the set referred to in the strong law of large numbers. Hint: Consider the map from $(0,1]$ to itself that maps the binary expression $\omega =. a_{1} a_{2} \ldots$ to. $a_{1} 11 a_{2} 11 a_{3} 11 a_{4} \ldots$ . Prove that this map is one-to-one and its image is contained in $(0,1] \backslash S$ .

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