Math Problems Vault

Problem 4.7.1

Construct a set in $\mathscr {L}^{k} - \mathscr {B}^{k}$ for each $k \geqslant 2$ .

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Remark.

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When $k \geqslant 2$ , there are very simple uncountable Borel sets of Lebesgue measure zero, namely, the hyperplanes. Use these and an appropriate homeomorphism to carry out the construction. Alternatively, one can find the answer given somewhat implicitly in Section 4.2.

Problem 4.7.2

The homeomorphic image of a Borel set is a Borel set.

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Problem 4.7.3

The homeomorphic image of a Lebesgue set is not necessarily a Lebesgue set. Therefore, the Lebesgue and Borel sets behave differently in this regard.

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Remark.

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From the construction, $\phi^{-1}(B) \in \mathscr {L}$ . The image of $\phi^{-1}(B)$ under the homeomorphism $\phi$ is $B \notin \mathscr {L}$ .