A theorem in measure theory stating that if (f_n) is an increasing sequence of nonnegative measurable functions that converges pointwise to f, then the integrals ∫ f_n converge to ∫ f (possibly infinite), i.e., lim_{n→∞} ∫ f_n = ∫ lim_{n→∞} f_n.

Principle

Monotonicity plus nonnegativity permits passage of the limit inside the integral because the integrals form an increasing sequence bounded above by the integral of the pointwise limit (or diverge together).

Demonstration

Take f_n = min(f, n) for a nonnegative measurable f; (f_n) increases to f, and by the theorem ∫ f_n ↗ ∫ f. Or let f_n = 1_{[0,1-1/n]} on R with Lebesgue measure: f_n ↑ 1_{[0,1)} and ∫ f_n → 1 = ∫ 1_{[0,1)}.

Misapplication

Applying the theorem to sequences that are not monotone increasing or not nonnegative (e.g., applying it to an alternating sequence) leads to false conclusions; monotonicity and nonnegativity are essential hypotheses.

Consequence

This theorem is a cornerstone for exchanging limits and integrals in many existence and approximation arguments; it implies Fatou's lemma and underpins simple-function approximations used to define the Lebesgue integral.

Reversal

The dominated convergence theorem gives a converse-type tool: if functions are dominated by an integrable function (instead of monotone), one can also exchange limit and integral, but the hypotheses and conclusions differ in direction and applicability.

Boundary

Requires measurable functions, monotone increasing pointwise almost everywhere, and nonnegativity. It does not apply to decreasing sequences (unless additional integrability assumptions hold) nor to sequences of signed functions without decomposition.

Semantic Tension

Tension with dominated convergence arises when monotonicity fails but domination holds; each theorem has trade-offs in hypotheses (monotone vs dominated) and in the ease of verifying conditions in applications.

Synthesis

The monotone convergence theorem gives a robust, limit-preserving rule for nonnegative monotone approximations: increasing measurable approximants converge in integral to the integral of their pointwise limit, forming a fundamental tool in Lebesgue integration theory.