Let fₙ = n · χ_{(0, 1/n)} on [0,1] with Lebesgue measure. The functions are non-negative and fₙ → 0 pointwise (a.e.). A student applies the MCT to conclude ∫fₙ dλ → 0. What is wrong?
ANothing — the MCT applies to any non-negative sequence converging pointwise, and the conclusion is correct
BThe sequence is not monotone increasing — MCT requires fₙ ≤ f_{n+1} pointwise, and this sequence fails that condition
CThe limit function f = 0 is not measurable, so MCT cannot be applied
DThe MCT only applies on finite measure spaces, and [0,1] must be extended to ℝ
The MCT requires 0 ≤ f₁ ≤ f₂ ≤ ... monotone pointwise. For this sequence, at x = 1/4: f₁(1/4)=1, f₂(1/4)=2, f₃(1/4)=3, f₄(1/4)=4, f₅(1/4)=0 — the sequence is NOT non-decreasing at every point. In fact ∫fₙ dλ = 1 for all n, so ∫fₙ dλ does not converge to 0. This is the classic counterexample showing why monotonicity is indispensable.
Question 2 Multiple Choice
The Lebesgue integral of a non-negative measurable function f is defined as ∫f dμ = lim ∫sₙ dμ, where sₙ ↑ f is an increasing sequence of simple functions approximating f from below. Why is the MCT essential to this definition?
AIt guarantees that every non-negative measurable function can be approximated by simple functions, which wouldn't otherwise be possible
BIt ensures the integral is always finite, which is required for the definition to make sense
CIt guarantees the limit lim ∫sₙ dμ is the same regardless of which approximating sequence is chosen, making the definition consistent
DIt proves that simple functions are dense in the space of non-negative measurable functions
The definition must be well-defined: ∫f dμ should be a property of f itself, not of a particular approximating sequence. Different choices of sₙ ↑ f yield different sequences of simple-function integrals, and the MCT guarantees they all converge to the same limit. Without this, the integral would be ambiguous. This is why MCT is the constructive engine of Lebesgue integration theory, not merely a convergence result.
Question 3 True / False
The Monotone Convergence Theorem guarantees that if 0 ≤ f₁ ≤ f₂ ≤ ... converges pointwise to f, then ∫fₙ dμ → ∫f dμ, and this limit is necessarily a finite real number.
TTrue
FFalse
Answer: False
The MCT allows the limit to be +∞. For example, fₙ = χ_{[0,n]} on ℝ with Lebesgue measure: the sequence is increasing, fₙ → χ_{[0,∞)} pointwise, and ∫fₙ dλ = n → ∞ = ∫χ_{[0,∞)} dλ. The MCT applies perfectly, and the conclusion is that both sides equal +∞. The theorem handles the infinite case gracefully — no finiteness assumption is needed.
Question 4 True / False
A monotone increasing sequence of measurable functions that are not non-negative can fail to satisfy ∫fₙ dμ → ∫f dμ even when fₙ → f pointwise.
TTrue
FFalse
Answer: True
Consider fₙ = −χ_{[n,∞)} on ℝ with Lebesgue measure. The sequence is increasing (fₙ(x) ≤ f_{n+1}(x) for all x) and fₙ → 0 pointwise as n → ∞. But ∫fₙ dλ = −∞ for every n, while ∫0 dλ = 0. The interchange fails catastrophically because mass is lost at infinity. This is why non-negativity is not a technical convenience — it prevents mass from 'escaping' to −∞ and destroying the convergence.
Question 5 Short Answer
Why do the non-negativity and monotone increasing conditions in the MCT prevent the 'mass escape' phenomenon that can cause the limit of integrals to differ from the integral of the limit?
Think about your answer, then reveal below.
Model answer: Non-negativity ensures integrals are non-negative real numbers (or +∞), so the sequence of integrals ∫fₙ dμ is a non-negative, non-decreasing sequence — it converges to a limit in [0, ∞] with no possibility of oscillation or loss. Monotonicity means each new function only adds measure-theoretic 'mass' rather than redistributing it: whatever area is captured under fₙ is still captured under f_{n+1}. This prevents mass from 'escaping' to regions where it wouldn't be counted in the limit, which is the failure mode in examples with negative functions or non-monotone sequences.
The key contrast is with unconstrained limits: a sequence can converge pointwise while concentrating increasing mass near infinity (or near a single point), causing the integrals to diverge even as the pointwise limit is zero (e.g., the 'moving bump' sequence). Monotonicity forbids mass from moving around; non-negativity forbids mass from canceling. Together they guarantee that the Lebesgue integral — defined by approximating from below — can be legitimately exchanged with the pointwise limit.