Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“A subset of ℝ with measure zero is countable” — false

Counterexample locked — unlock with Notes + PYQ

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#2 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“A set of measure zero is countable” — false

Counterexample: The Cantor set

Uncountable, yet measure zero.

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#3 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“A nowhere dense set has measure zero” — false

Counterexample locked — unlock with Notes + PYQ

measure

#4 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“Pointwise convergence implies convergence of the integrals” — false

Counterexample: fₙ = n·1_{(0,1/n)} on [0,1]

$f_{n} \to 0$ pointwise but $\int f_{n} = 1$ always. Domination or monotonicity is essential.

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#5 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“Iterated integrals of a measurable function always agree” — false

Counterexample locked — unlock with Notes + PYQ

measure

#6 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“L¹[0,1] ⊆ L²[0,1]” — false

Counterexample: f(x) = 1/√x

$\int f = 2 < \infty$ but $\int f^{2} = \int$ dx $/ x = \infty$ . On a finite measure space the inclusion runs the other way.

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