Squeeze-style lemma in ℝ≥0∞: if b n ≤ a n, a n → c < ∞, and the liminf of a n - b n
is at least c, then b n → 0.
Pointwise (a.e.) convergence step in Scheffé's lemma: if f n → g a.e., then the auxiliary
quantity ‖f n‖ₑ + ‖g‖ₑ - ‖f n - g‖ₑ converges a.e. to 2‖g‖ₑ.
Integration identity used in the proof of Scheffé's lemma: the lintegral of the auxiliary
quantity equals (eLpNorm (f n) 1 μ + eLpNorm g 1 μ) - eLpNorm (f n - g) 1 μ.
Scheffé's lemma. If f n → g almost everywhere and ‖f n‖₁ → ‖g‖₁, then f n → g in
L¹, i.e. eLpNorm (f n - g) 1 μ → 0.
If f n → g in L¹ (i.e. eLpNorm (f n - g) 1 μ → 0), then the L¹-norms converge:
‖f n‖₁ → ‖g‖₁.
If f n → g in L¹ and g, f n are all in L¹, then the eLpNorms of f n are uniformly
bounded by some C : ℝ≥0.
L¹ convergence implies uniform integrability: if f n → g in L¹ (with f n, g in L¹),
then the family f n is uniformly integrable in L¹.
L¹ convergence f n → g implies convergence of the L¹ norms ‖f n‖₁ → ‖g‖₁. This is
the (2) ⇒ (3) implication of the uniform integrability equivalence theorem.
If f n → g in measure and ‖f n‖₁ → ‖g‖₁, then the family f n is uniformly integrable
in L¹. This is the (3) ⇒ (1) implication of the uniform integrability equivalence theorem,
proven via Scheffé's lemma and a subsequence argument.
If f n is uniformly integrable in L¹ and f n → g in measure (with g ∈ L¹), then
f n → g in L¹. This is the (1) ⇒ (2) implication of the uniform integrability equivalence
theorem.
Uniform integrability equivalences. Suppose f n → g in measure on a finite measure
space, with f n, g ∈ L¹. Then the following are equivalent:
- The family
f nis uniformly integrable inL¹. f n → ginL¹(i.e.eLpNorm (f n - g) 1 μ → 0).‖f n‖₁ → ‖g‖₁.
Renamed alias for ui_L1_norm_tfae: the textbook statement that uniform integrability,
L¹ convergence, and convergence of L¹ norms are equivalent (assuming convergence in
measure to g).
Top-level (out-of-namespace) restatement of the uniform integrability equivalences:
if f n → g in measure on a finite measure space with f n, g ∈ L¹, then uniform integrability
of f n, convergence f n → g in L¹, and convergence ‖f n‖₁ → ‖g‖₁ are equivalent.
Out-of-namespace alias for TheoryOfProbability3.ui_L1_norm_tfae, stating the same
uniform integrability ↔ L¹ convergence ↔ convergence of L¹ norms equivalence.