Let Xₙ = n · 𝟏{0 < U < 1/n} where U ~ Uniform(0,1). Then Xₙ → 0 almost surely, yet E[Xₙ] = 1 for all n. Which theorem fails to apply here, and why?
AThe Monotone Convergence Theorem — it fails because the sequence is not monotone increasing
BThe Dominated Convergence Theorem — it fails because there is no integrable function g with |Xₙ| ≤ g for all n
CThe Law of Large Numbers — it fails because the Xₙ are not identically distributed
DBoth MCT and DCT — they fail because Xₙ does not converge in L¹
The DCT would allow E[lim Xₙ] = lim E[Xₙ] if there existed an integrable dominating function g with |Xₙ(ω)| ≤ g(ω) for all n and almost all ω. But Xₙ = n on (0, 1/n), and the supremum sup_n Xₙ = ∞ on (0,1) — any candidate g would need to be infinite on this set, which is not integrable. Without a dominating function, the DCT does not apply, and the exchange of limit and expectation fails. This is the canonical example showing why the dominated convergence condition is not merely technical — it is necessary.
Question 2 Multiple Choice
What is the key advantage of defining expectation as E[X] = ∫_Ω X dP (a Lebesgue integral on the probability space) over the elementary definitions E[X] = Σ xᵢP(X = xᵢ) or E[X] = ∫ x f(x) dx?
AThe measure-theoretic definition is easier to compute numerically for most practical distributions
BIt provides a single unified framework that handles discrete, continuous, and mixed distributions — and distributions with no density — under one definition
CIt automatically guarantees that every random variable has a finite expectation
DIt eliminates the need to check measurability conditions for the random variable
The elementary formulas only work in their respective special cases: the sum requires a discrete distribution, the integral requires an absolutely continuous one with a density. The Lebesgue integral E[X] = ∫_Ω X dP works for any random variable on any probability space — discrete, continuous, singular (like the Cantor distribution), or mixed. This generality is not just aesthetically satisfying; it is necessary for the general theory of conditional expectation, convergence theorems, and stochastic processes, none of which can be built cleanly on the elementary definitions alone.
Question 3 True / False
If Xₙ ≥ 0 for all n and Xₙ increases pointwise to X (possibly with X = ∞ on some set), the Monotone Convergence Theorem guarantees that E[Xₙ] → E[X], even when E[X] = ∞.
TTrue
FFalse
Answer: True
The MCT holds without requiring finiteness of the limit: if 0 ≤ X₁ ≤ X₂ ≤ ⋯ and Xₙ → X pointwise, then ∫ Xₙ dP → ∫ X dP, where both sides may equal +∞. This is a strength of the Lebesgue integral — it handles the infinite case cleanly. The conclusion 'E[Xₙ] → ∞' is meaningful and correct when E[X] = ∞. In contrast, DCT requires a finite dominating integrable function, so DCT cannot handle this infinite limit case.
Question 4 True / False
Nearly every random variable defined on a probability space has a well-defined finite expectation, since probabilities are bounded between 0 and 1.
TTrue
FFalse
Answer: False
Boundedness of P (a probability measure) does not imply that integrals of X are finite. A random variable can take arbitrarily large values with just enough probability that ∫ |X| dP = ∞. The Cauchy distribution is the canonical example: its density is f(x) = 1/(π(1+x²)), symmetric about zero, but ∫ |x| f(x) dx diverges. For the Cauchy distribution, E[X] is undefined — neither finite nor infinite in a well-defined sense. Integrability (E[|X|] < ∞) must always be verified, not assumed.
Question 5 Short Answer
Why must we verify that E[|X|] < ∞ (absolute integrability) rather than just checking that ∫_Ω X dP converges, before concluding that E[X] is well-defined?
Think about your answer, then reveal below.
Model answer: The Lebesgue integral ∫ X dP is defined as ∫ X⁺ dP − ∫ X⁻ dP, where X⁺ and X⁻ are the positive and negative parts of X. If both integrals are finite, E[X] is well-defined. But if both ∫ X⁺ dP = ∞ and ∫ X⁻ dP = ∞, then E[X] = ∞ − ∞ is undefined — not a number. Checking E[|X|] = ∫ X⁺ dP + ∫ X⁻ dP < ∞ ensures both parts are finite, so their difference is a well-defined real number.
This is analogous to the distinction between absolutely convergent and conditionally convergent series: a conditionally convergent series can be rearranged to converge to any value or diverge, while an absolutely convergent series has a unique well-defined sum. The Lebesgue integral is by design an 'absolute' integral — it does not extend naturally to conditionally convergent situations. Requiring E[|X|] < ∞ is what puts X in L¹(P), the natural function space for expectation, and is the prerequisite for most convergence theorems (DCT, uniform integrability, etc.).