On master test plans for the space of BV functions

Definition A.1 (Approximation modulus [31]).

Let ( X , 𝖽 , 𝔪 ) be a metric measure space and let Ω ⊆ X be an open set. Let Γ ⊆ 𝒞 ⁢ ( Ω ) be a given family of curves. Then a sequence ( ρ i ) i ∈ ℕ of non-negative Borel functions ρ i : Ω → ℝ is said to be AM Ω -admissible for Γ provided it holds

The approximation modulus of Γ in Ω is defined by

where the infimum is taken among all AM Ω -admissible sequences ( ρ i ) i for Γ. Moreover, a property 𝒫 = 𝒫 ⁢ ( γ ) is said to hold for AM Ω -a.e. curve γ provided there exists a family of curves Γ 0 ⊆ 𝒞 ⁢ ( Ω ) with AM Ω ⁡ ( Γ 0 ) = 0 such that 𝒫 ⁢ ( γ ) holds for every γ ∈ 𝒞 ⁢ ( Ω ) ∖ Γ 0 .

Lemma A.2.

Let ( X , d , m ) be a metric measure space. Let Γ ⊆ C ⁢ ( [ 0 , 1 ] , X ) be a family of curves such that AM ⁡ ( Γ ) = 0 . Fix an ∞ -test plan 𝛑 on ( X , d , m ) . Then there exists a Borel set Γ 0 ⊆ C ⁢ ( [ 0 , 1 ] , X ) such that Γ ⊆ Γ 0 and 𝛑 ⁢ ( Γ 0 ) = 0 .

Proof.

Given n ∈ ℕ , pick an AM-admissible sequence ( ρ i n ) i for Γ such that

Let us consider the Borel sets

Then the set Γ 0 ≔ ⋂ n Γ n is Borel and contains Γ. We aim to show that 𝝅 ⁢ ( Γ 0 ) = 0 . Since

holds for every n ∈ ℕ , by letting n → ∞ , we conclude that 𝝅 ⁢ ( Γ 0 ) = 0 , as desired. ∎

Remark A.3.

Let Γ , Γ ′ ⊆ 𝒞 ⁢ ( X ) be two families of curves having the following property: given any γ ∈ Γ , some subcurve σ of γ belongs to Γ ′ , meaning that there exists a compact subinterval I of I γ such that σ ≔ γ | I ∈ Γ ′ . Then it holds that AM ⁡ ( Γ ) ≤ AM ⁡ ( Γ ′ ) .

Definition A.4 ( BV AM upper bound [32]).

Let ( X , 𝖽 , 𝔪 ) be a metric measure space and let Ω ⊆ X be an open set. Let f ∈ ℒ 1 ( 𝔪 | Ω ) be given. Then we say that a sequence ( g i ) i ∈ ℕ of non-negative Borel functions g i : Ω → ℝ is a BV AM upper bound for f on Ω provided it holds that

Definition A.5 (AM-BV space [32]).

Let ( X , 𝖽 , 𝔪 ) be a metric measure space. Fix a function f ∈ L 1 ⁢ ( 𝔪 ) . Then we say that f belongs to the space BV AM ⁡ ( X ) provided there exist a representative f ¯ ∈ ℒ 1 ⁢ ( 𝔪 ) of f and a BV AM upper bound ( g i ) i for f ¯ such that

Given any f ∈ BV AM ⁡ ( X ) and Ω ⊆ X open, we define

where the infimum is taken among all representatives f ¯ ∈ ℒ 1 ⁢ ( 𝔪 ) of f and all BV AM upper bounds ( g i ) i for f ¯ on Ω.

Theorem A.6 ( BV AM ⁡ ( X ) = BV ⁡ ( X ) ).

Let ( X , d , m ) be a metric measure space. Then it holds

Proof.

By virtue of Theorem 4.3, it suffices to show BV ⁡ ( X ) ⊆ BV AM ⁡ ( X ) ⊆ BV 𝖼𝗐 ⁡ ( X ) and

We will prove it in two steps.

Step 1. First, we want to prove that BV ⁡ ( X ) ⊆ BV AM ⁡ ( X ) and that | 𝐃 ⁢ f | AM ⁢ ( Ω ) ≤ | 𝐃 ⁢ f | ⁢ ( Ω ) for every f ∈ BV ⁡ ( X ) and Ω ⊆ X open. Thanks to Theorem 2.10, we can find a sequence ( f i ) i ⊆ LIP loc ( Ω ) ∩ L 1 ( 𝔪 | Ω ) such that f i → f in L 1 ( 𝔪 | Ω ) and ∫ Ω lip a ⁡ ( f i ) ⁢ d 𝔪 → | 𝐃 ⁢ f | ⁢ ( Ω ) . Fix a representative f ¯ ∈ ℒ 1 ( 𝔪 | Ω ) of f. It follows from Fuglede’s lemma [14, Lemma 2.1] that (up to a not relabelled subsequence) it holds that ∫ γ | f i - f ¯ | → 0 as i → ∞ for AM Ω -a.e. γ. In particular, we have that f i ∘ γ → f ¯ ∘ γ strongly in L 1 ⁢ ( 0 , 1 ) for AM Ω -a.e. γ having constant speed. By using the lower semicontinuity of the total variation measures, we thus obtain that

This shows that ( lip a ⁡ ( f i ) ) i is a BV AM upper bound for f ¯ on Ω. Therefore, we conclude that

as desired.

Step 2. Next, we aim to prove that BV AM ⁡ ( X ) ⊆ BV 𝖼𝗐 ⁡ ( X ) and that | 𝐃 ⁢ f | 𝖼𝗐 ⁢ ( Ω ) ≤ | 𝐃 ⁢ f | AM ⁢ ( Ω ) for every f ∈ BV AM ⁡ ( X ) and Ω ⊆ X open. Given any ε > 0 , pick a representative f ¯ ∈ ℒ 1 ⁢ ( 𝔪 ) of f and a BV AM upper bound ( g i ) i for f ¯ on Ω such that lim ¯ i ⁡ ∫ Ω g i ⁢ d 𝔪 ≤ | 𝐃 ⁢ f | AM ⁢ ( Ω ) + ε . Fix a family Γ ⊆ 𝒞 ⁢ ( Ω ) such that AM ⁡ ( Γ ) = 0 and (A.3) holds for all γ ∉ Γ . In light of Remark A.3, we can also assume without loss of generality that if σ ∈ Γ , then any curve γ ∈ 𝒞 ⁢ ( X ) having σ as a subcurve belongs to Γ. Now, let 𝝅 be a given ∞ -test plan on ( X , 𝖽 , 𝔪 ) . By applying Lemma A.2, we can find a Borel set Γ 0 ⊆ C ⁢ ( [ 0 , 1 ] , X ) such that Γ ∩ C ⁢ ( [ 0 , 1 ] , X ) ⊆ Γ 0 and 𝝅 ⁢ ( Γ 0 ) = 0 . Now, fix γ ∈ LIP ⁡ ( [ 0 , 1 ] , X ) ∖ Γ 0 and 0 < a < b < 1 with γ ⁢ ( ( a , b ) ) ⊆ Ω . Given that γ ∉ Γ , we have that γ | [ a , b ] ∉ Γ as well. Thus accordingly,

This shows that ( g i ) i is a curvewise bound for f on Ω. In particular, we deduce that

whence, by letting ε ↘ 0 , we conclude that | 𝐃 ⁢ f | 𝖼𝗐 ⁢ ( Ω ) ≤ | 𝐃 ⁢ f | AM ⁢ ( Ω ) , as desired. ∎

Definition B.1 (The space W 1 , 1 ⁢ ( X ) ).

Let ( X , 𝖽 , 𝔪 ) be a metric measure space. We say that f ∈ W 1 , 1 ⁢ ( X ) provided f ∈ L 1 ⁢ ( 𝔪 ) and there exists G ∈ L 1 ⁢ ( 𝔪 ) non-negative, called 1-weak upper gradient of f, so that

for every ∞ -test plan 𝝅 .

The 𝔪 -a.e. minimal G satisfying the above, denoted by | D ⁢ f | 1 , is called minimal 1-weak upper gradient.

Theorem B.2.

Let ( X , d , m ) be an RCD ⁢ ( K , N ) space with N < ∞ and m finite. Then there exists an ∞ -test plan, denoted by π m and concentrated on geodesics, so that if f , G ∈ L 1 ⁢ ( m ) are such that f ∘ γ ∈ W 1 , 1 ⁢ ( 0 , 1 ) for 𝛑 m -a.e. γ ∈ AC ⁡ ( [ 0 , 1 ] , X ) and

then f ∈ W 1 , 1 ⁢ ( X ) and G is a 1-weak upper gradient.

Proof.

Since 𝖱𝖢𝖣 ⁢ ( K , N ) spaces are non-branching [19, Theorem 1.3], we know from Theorem 3.10 that we can find a countable collection Π of ∞ -test plans concentrated on geodesics which is a master family for BV ⁡ ( X ) . An argument as in the proof of Theorem 4.6 gives rise to, out of the countable collection Π, a single ∞ -test plan 𝝅 𝗆 which is concentrated on geodesics with length at most 1 and satisfying the key property:

for every Borel Γ ⊆ C ⁢ ( [ 0 , 1 ] , X ) . Finally, f , G ∈ L 1 ⁢ ( 𝔪 ) satisfy (B.2) if and only if

This implies that for 𝝅 -a.e. γ it holds that f ∘ γ ∈ BV ⁡ ( 0 , 1 ) (in fact, it is absolutely continuous) with

for every I ⊆ [ 0 , 1 ] Borel and 𝝅 ∈ Π . Thus, we reach

for every B ⊆ X Borel and 𝝅 ∈ Π . This means that f ∈ BV Π ⁡ ( X ) and | 𝐃 ⁢ f | Π ≤ G ⁢ 𝔪 . By Theorem 3.10, this immediately implies that f ∈ BV ⁡ ( X ) with | 𝐃 ⁢ f | ≤ G ⁢ 𝔪 and, by recalling (B.1), also the conclusion. ∎