Extended non-negative reals #

We define ennreal = ℝ≥0∞ := with_top ℝ≥0 to be the type of extended nonnegative real numbers, i.e., the interval [0, +∞]. This type is used as the codomain of a measure_theory.measure, and of the extended distance edist in a emetric_space. In this file we define some algebraic operations and a linear order on ℝ≥0∞ and prove basic properties of these operations, order, and conversions to/from ℝ, ℝ≥0, and ℕ.

Main definitions #

• ℝ≥0∞: the extended nonnegative real numbers [0, ∞]; defined as with_top ℝ≥0; it is equipped with the following structures:

  • coercion from ℝ≥0 defined in the natural way;

  • the natural structure of a complete dense linear order: ↑p ≤ ↑q ↔ p ≤ q and ∀ a, a ≤ ∞;

  • a + b is defined so that ↑p + ↑q = ↑(p + q) for (p q : ℝ≥0) and a + ∞ = ∞ + a = ∞;

  • a * b is defined so that ↑p * ↑q = ↑(p * q) for (p q : ℝ≥0), 0 * ∞ = ∞ * 0 = 0, and a * ∞ = ∞ * a = ∞ for a ≠ 0;

  • a - b is defined as the minimal d such that a ≤ d + b; this way we have ↑p - ↑q = ↑(p - q), ∞ - ↑p = ∞, ↑p - ∞ = ∞ - ∞ = 0; note that there is no negation, only subtraction;

  • a⁻¹ is defined as Inf {b | 1 ≤ a * b}. This way we have (↑p)⁻¹ = ↑(p⁻¹) for p : ℝ≥0, p ≠ 0, 0⁻¹ = ∞, and ∞⁻¹ = 0.

  • a / b is defined as a * b⁻¹.

  The addition and multiplication defined this way together with 0 = ↑0 and 1 = ↑1 turn ℝ≥0∞ into a canonically ordered commutative semiring of characteristic zero.

• Coercions to/from other types:

  • coercion ℝ≥0 → ℝ≥0∞ is defined as has_coe, so one can use (p : ℝ≥0) in a context that expects a : ℝ≥0∞, and Lean will apply coe automatically;

  • ennreal.to_nnreal sends ↑p to p and ∞ to 0;

  • ennreal.to_real := coe ∘ ennreal.to_nnreal sends ↑p, p : ℝ≥0 to (↑p : ℝ) and ∞ to 0;

  • ennreal.of_real := coe ∘ real.to_nnreal sends x : ℝ to ↑⟨max x 0, _⟩

  • ennreal.ne_top_equiv_nnreal is an equivalence between {a : ℝ≥0∞ // a ≠ 0} and ℝ≥0.

Implementation notes #

We define a can_lift ℝ≥0∞ ℝ≥0 instance, so one of the ways to prove theorems about an ℝ≥0∞ number a is to consider the cases a = ∞ and a ≠ ∞, and use the tactic lift a to ℝ≥0 using ha in the second case. This instance is even more useful if one already has ha : a ≠ ∞ in the context, or if we have (f : α → ℝ≥0∞) (hf : ∀ x, f x ≠ ∞).

Notations #

• ℝ≥0∞: the type of the extended nonnegative real numbers;
• ℝ≥0: the type of nonnegative real numbers [0, ∞); defined in data.real.nnreal;
• ∞: a localized notation in ℝ≥0∞ for ⊤ : ℝ≥0∞.