Phase 6 Exam

Real Analysis, Measure & Integration - Phase Exam

15 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches

Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.

Q1 Medium

Why is \(\mathbb{Q}\) not complete?

It is uncountable

It has a Cauchy sequence (e.g. approaching \(\sqrt2\)) whose limit is not rational

It is not bounded

It has no open sets

Q2 Medium

Which property must a sigma-algebra satisfy that a mere algebra of sets need not?

Closure under complement

Containing \(X\)

Closure under COUNTABLE unions

Being nonempty

Q3 Easy

The essential difference between the Lebesgue and Riemann integrals is that Lebesgue:

Partitions the range (y-axis) and weights level sets by measure

Partitions the domain into finer subintervals

Only works for continuous functions

Always gives a larger value

Q4 Medium

For \(f_n=n\mathbf{1}_{(0,1/n)}\) on \([0,1]\), which statement is correct?

\(\lim\int f_n=\int\lim f_n=1\)

\(\lim\int f_n=1\) but \(\int\lim f_n=0\)

Both limits are 0

DCT applies and gives equality

Q5 Easy

The expectation \(\mathbb{E}[X]\) is, in measure-theoretic terms:

A derivative of the distribution

The Lebesgue integral \(\int_\Omega X\,d\mathbb{P}\)

The supremum of \(X\)

A product measure

Q6 Easy

In \(\mathbb{R}^n\), a set is compact if and only if it is:

Open and bounded

Closed and bounded

Closed and connected

Bounded and countable

Q7 Medium

Countable additivity of a measure directly implies all of the following EXCEPT:

Monotonicity

Continuity from below

That every set is measurable

Countable subadditivity

Q8 Medium

The Dirichlet function \(\mathbf{1}_{\mathbb{Q}}\) on \([0,1]\) has Lebesgue integral:

Undefined

Infinite

Q9 Easy

Which theorem gives a one-sided inequality valid with NO domination or monotonicity hypothesis?

Monotone convergence

Fatou's lemma

Dominated convergence

None; all need extra hypotheses

Q10 Medium

Hölder’s inequality at \(p=q=2\) specializes to:

Minkowski’s inequality

The triangle inequality

Cauchy–Schwarz

Jensen’s inequality

Q11 Medium

A continuous function on a compact set is guaranteed to:

Be differentiable

Attain its maximum and minimum

Be linear

Have no zeros

Q12 Medium

The Cantor set is an example of a set that is:

Countable and of positive measure

Uncountable and of measure zero

Finite

Non-measurable

Q13 Medium

An \(f\) is in \(L^1(\mu)\) precisely when:

\(f\) is continuous

\(\int|f|\,d\mu\lt \infty\)

\(f\) is bounded

\(\int f\,d\mu=0\)

Q14 Medium

To apply the dominated convergence theorem you must exhibit:

A monotone increasing sequence

An integrable \(g\) with \(|f_n|\le g\) for all \(n\)

Continuity of each \(f_n\)

A finite measure space

Q15 Medium

Fubini’s theorem (as opposed to Tonelli) additionally requires:

\(f\ge0\)

\(f\in L^1(\mu\times\nu)\)

A finite measure space

Continuity of \(f\)

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