A basic inequality in measure theory: for a sequence (f_n) of nonnegative measurable functions, the integral of the pointwise liminf is less than or equal to the liminf of the integrals, i.e. ∫ liminf_{n} f_n ≤ liminf_{n} ∫ f_n.

Principle

Lower semicontinuity of the integral under liminf: nonnegativity prevents cancellation and ensures that mass lost in pointwise limits cannot force the liminf of integrals below the integral of the liminf.

Demonstration

Let f_n(x) = n·1_{[0,1/n]} on R with Lebesgue measure. Then f_n → 0 pointwise, so ∫ liminf f_n = 0, while each ∫ f_n = 1, so 0 ≤ liminf ∫ f_n = 1, illustrating the inequality and the possibility of strictness.

Misapplication

Attempting to apply Fatou's lemma to sequences that are not nonnegative (signed functions) without splitting into positive and negative parts, or wrongly replacing liminf by limsup; both lead to incorrect inequalities.

Consequence

Fatou's lemma is a key step in proving the monotone and dominated convergence theorems and provides a minimal lower bound control on limiting integrals, used broadly in existence proofs and variational arguments.

Reversal

There is no general reverse inequality; to obtain equality or an upper bound one typically requires stronger hypotheses such as dominated convergence or uniform integrability, which control the loss of mass.

Boundary

Valid only for nonnegative measurable functions (or applied to nonnegative parts of signed functions). It applies in general measure spaces but does not by itself yield upper bounds or convergence without additional hypotheses.

Semantic Tension

Tension appears with dominated convergence (which gives limit-exchange under domination) and monotone convergence (which gives equality under monotonicity); Fatou provides a minimal inequality that other theorems refine under stronger conditions.

Synthesis

Fatou's lemma furnishes a robust lower semicontinuity statement for integrals under pointwise liminf of nonnegative functions: it guarantees that integral mass does not disappear below the liminf of integrals, forming a foundational inequality in integration theory.