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Is this true?

A nowhere dense set has measure zero

No — it is false.

The counterexample

This one comes with the Notes pack.

See pricing

85 counterexamples are free — including most of the ones Part C leans on hardest.

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this
measure

Others that fail the same way

From Lebesgue Measure and Integration › Measurable sets and functions

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