Two measure-free versions of the Brezis - Lieb lemma

The Brezis–Lieb lemma [2, Theorem 2] has numerous applications mainly in calculus of variations (see, for example [3, 6]). We begin with its statement. Let j : C A C be a continuous function with j(0) = 0. In addition, let j satisfy the following hypothesis: for every sufficiently small ε > 0, there exist two continuous, nonnegative functions ϕ _ε and ψ _ε such that

^{| j} (a + b) ^{- j(a) |} 6 E^ _e ^(a) + <^{(b)                             (1)}

for all a,b E C. The following result has been stated and proved by H. Brezis and E. Lieb in [2].

Theorem 1.1 (Brezis–Lieb lemma [2, Theorem 2]) . Let (Ω , Σ , µ ) be a measure space. Let the mapping j satisfy the above hypothesis, and let f _n = f + g _n be a sequence of measurable functions from Ω to C such that:

• (i)    g n -A 0;

• (ii)    j ° f E L ¹ ;

• (iii)    J y _e ° g n dp 6 C <  ж for some C independent of e and n;

• (iv)    J fy ° fdp < ж for all e >  0 .

Then, as n A ж , j(j(f + g n ) - j(g n ) - j(f )W a 0-                       (2)

Here we reproduce its proof from [2, Theorem 2] with several simple remarks.

C Fix e > 0 and let W _e ,n = [| j ° f n - j ° g n - j ° f | - e^ _£ ° g n ]₊. As n A ж, W _e ,n --A 0. On the other hand,

|j ° f n - j ° g n - j ° f | 6 |j ° f n - j ° g n | + |j ° f | 6 e^ _£ ° g n + < ° f + |j ° f |.

Therefore, 0 6 Ws,n 6 ^e о f + |j о f | g L1. By dominated convergence, lim n→∞

/

W ε,n dµ = 0.

However,

|j о f n - j о g n — j о f | 6 W _s,n + Е/ о g n                      (4)

and thus

I n ^:

= j j о fn

- j о g n

-

j ^{о f l d}^ 6 j

[W _e ,n + E^ _e о g n ]d^.

Consequently, limsup I n 6 eC . Now let e 4 0. B

Remark 1.1. (i) The conditions (3) and (4) mean that the sequence |j о f _n — j о g _n | lies eventually in the set [—|j о f |, |j о f |] + ^C Bl i , where Bl i is the unit ball of L 1 . In other words, the sequence j о f n — j о g _n is almost order bounded.

• (ii)    The superposition operator J j : L0 4 L0, J j (f ) := j о f induced by the mapping j in the proof above can be replaced by a mapping J : L° 4 L° satisfying some reasonably mild conditions for keeping the statement of the Brezis–Lieb lemma.

• (iii)    Theorem 1.1 is equivalent to its partial case when the C-valued functions are replaced by R-valued ones.

The following proposition is motivated directly by the proof of [2, Theorem 2].

Proposition 1.2 (Brezis–Lieb lemma for mappings on L0). Let (Ω, Σ, µ) be a measure space, fn = f + gn be a sequence in L° such that gn -—4 0, and J : L° 4 L° be a mapping satisfying J(0) = 0 and such that the sequence J(fn) — J(gn) is almost order bounded. Then lim n→∞

/

( ^{J(f +} g n ) ^{— (J(g} n ) ^{+ J ( f} )) ) ^d ^

= 0 .

C As in the proof of the Brezis-Lieb lemma above, denote In := J | J(f + g _n ) — (J(f) + J(g n ))| d^. By the conditions, the sequence

J (f + g n ) — (J (f) + J (g n )) = (J (f n ) — J (g n ))) — J (f )

a.e.-converges to 0 and is almost order bounded. Therefore, by the generalized dominated convergence, lim In = 0. B n→∞

Since almost order boundedness is equivalent to uniform integrability in finite measure spaces, the following corollary is immediate.

Proposition 1.3 (Brezis-Lieb lemma for uniform integrable sequence J(fn) — J(gn)). Let (Q, X, ^) be a finite measure space, fn = f + gn be a sequence in L° such that gn —4 0, and J : L0 4 L0 be a mapping satisfying J(0) = 0 and such that the sequence J(fn) — J(gn) is uniformly integrable. Then lim n→∞

I (J (f + g n ) — (J (g n ) + J (f )))d^

=0 .

• 2.    Two variants of the Brezis–Lieb lemma in Riesz spaces

Recall that a sequence x _n in a Riesz space E is order convergent (or o-convergent, for short) to x ∈ E if there is a sequence z _n in E satisfying z _n ↓ 0 and |x _n - x| 6 z _n for all n ∈ N (we write x _n -→ ^o x). In a Riesz space E, a sequence x _n is unbounded order convergent (or uo-convergent, for short) to x ∈ E if |x _n - x| ∧ y -→ ^o 0 for all y ∈ E+ (we write x _n - ^u → ^o x).

Here we give two variants of the Brezis–Lieb lemma in Riesz space setting by replacing a.e.-convergence by uo-convergence, integral functionals by strictly positive functionals and the continuity of the scalar function j (in Theorem 1.1) by the so called σ-unbounded order continuity of the mapping J : E → F between Riesz spaces E and F . As standard references for basic notions on Riesz spaces we adopt the books [1, 7, 8] and on unbounded order convergence the papers [4, 5].

It is well known that if (Ω, Σ, µ) be a σ-finite measure space, then in L ^p (1 6 p 6 ∞), uo-convergence of sequences is the same as the almost everywhere convergence (see, for example [5]). Therefore, in order to obtain versions of Brezis–Lieb lemma in Riesz spaces, we shall replace the a.e.-convergence by the uo-convergence.

A mapping f : E → F between Riesz spaces is said to be σ-unbounded order continuous (in short, σuo-continuous) if x _n - ^u → ^o x in E implies f (x _n ) - ^u → ^o f (x) in F . Clearly this definition is parallel to the well-known notion of σ-order continuous mappings between Riesz spaces.

Let F be a Riesz space and l be a strictly positive linear functional on F . Define the following norm on F :

kx k l := l(|x|).                                                       (7)

Recall that a Banach lattice E is said to be order continuous if every order null net is norm null, and a subset A of E is said to be almost order bounded if for any ε > 0 there exists u _ε ∈ E+ such that A ⊂ [-u _ε , u _ε ] + εB E , where B E is the closed unit ball in E. We say that a net x _α is almost order bounded if the set of its members is almost order bounded.

The next lemma will be used to prove a version of Brezis–Lieb lemma for arbitrary strictly positive linear functionals.

Lemma 2.1 (See [5, Proposition 3.7]) . Let X be an order continuous Banach lattice. If a net x _α is almost order bounded and uo -convergent to x , then x _α converges to x in norm.

Suppose that F is a Riesz space and l is a strictly positive linear functional on F , then the k · k l -completion (F l , k . k l ) of (F, k · k l ) is an AL-space, and so it is order continuous Banach lattice. The following result is a measure-free version of Proposition 1.2.

Proposition 2.2 (A Brezis–Lieb lemma for strictly positive linear functionals) . Let E be a Riesz space and F l be the AL -space constructed above. Let J : E → F l be σuo -continuous with J(0) = 0 , and x _n be a sequence in E such that:

• (i)    x _n - ^u → ^o x in E ;

• (ii)    the sequence (J (x _n ) - J(x _n - x)) _n is almost order bounded in F _l .

Then lim k J(xn) - J(xn - x) - J(x) kl= 0.                          (8)

n→∞

C Since x _n - ^u → ^o x and J is σuo-continuous, then J(x _n ) - ^u → ^o J(x) and J(x _n - x) - ^u → ^o J(0) = 0. Thus, J(x _n ) - J(x _n - x) - ^u → ^o J (x). It follows from Lemma 2.1 that lim k J(x n ) - J(x n - x) - J(x) k l = 0. B n→∞

In the following Brezis–Lieb type lemma, the σuo-continuity of mappings between Riesz spaces is used.

Proposition 2.3 (A Brezis–Lieb lemma for σuo-continuous linear functionals). Let E, F be Riesz spaces, l a σuo-continuous linear functional on F , J : E → F a σuo-continuous mapping with J (0) = 0, and xn -u→o x in E. Then lim l(J (xn) - J(xn - x) - J (x)) = 0 .                          (9)

n→∞

C Since x _n - ^u → ^o x and J is σuo-continuous, then J(x _n ) - ^u → ^o J(x) and J(x _n - x) - ^u → ^o J(0) = 0. Thus, (J(x _n ) - J(x _n - x) - J (x)) - ^u → ^o 0. But l is σuo-continuous, so l(J (x _n ) - J(x _n - x) - J (x)) - ^u → ^o 0. Since in R the uo-convergence, the o-convergence, and the standard convergence are all equivalent, then lim l(J (x _n ) - J(x _n - x) - J (x)) = 0. B n→∞

Note that in opposite to Proposition 2.3, in Proposition 2.2 we do not suppose the functional l to be σuo-continuous.