On a theorem of Baire about lower semi-continuous functions

Costica Mustata
(original), Approximation Theory, paper

Abstract


A theorem of Baire concerning the approximation of lower semicontinuous real valued functions defined on a metric space, by increasing sequences of continuous functions is extended to the “nonsymmetric” case, i.e. for quasi-metric spaces.

Authors

Costică Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis , Cluj-Napoca, Romania

Keywords

Quasi-metric space, semi-Lipschitz function, approximation.

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C. Mustăţa, On a theorem of Baire about lower semi-continuous functions, Rev. Anal. Numer. Theor. Approx. 37 (2008) no. 1, 71-75.

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Revue d’Analyse Numer Theorie Approximation

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Publisher House of the Romanian Academy

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https://ictp.acad.ro/jnaat/journal/article/view/2008-vol37-no1-art8

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[1] Baire, R.,Lecon sur les fonctions discontinues,Paris, Collection Borel, 1905, pp. 121–123.
[2] Borodin, P.A.,The Banach-Mazur theorem for spaces with asymmetric norm and itsapplications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.
[3] Collins, J.and  Zimmer, J.An asymmetric Arzela-Ascoli theorem,Topology Appl.,154, no. 11, pp. 2312–2322, 2007.
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[5] McShane, E.T., Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 837–842, 1934.
[6] Menucci, A., On asymmetric distances,Technical Report, Scuola Normale Superiore,Pisa, 2004.
[7] Mustata, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[8] Nicolescu, M., Mathematical Analysis, Vol. II, Editura Tehnica, Bucharest, p. 119,1958 (in Romanian).
[9] Precupanu, A., Mathematical Analysis: Measure and Integration, Vol. I., EdituraUniversitatii A.I. Cuza Iasi, 2006 (Romanian).
[10] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best-approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[11] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitzfunctions. Acta Math. Hungar.,108, no. 1-2, pp. 55–70, 2005.

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2008-Mustata-ON A THEOREM OF BAIRE-Jnaat

ON A THEOREM OF BAIRE ABOUT LOWER SEMICONTINUOUS FUNCTIONS*

COSTICĂ MUSTĂŢA † † ^(†){ }^{\dagger}†

Abstract

A theorem of Baire concerning the approximation of lower semicontinuous real valued functions defined on a metric space, by increasing sequences of continuous functions is extended to the "nonsymmetric" case, i.e. for quasimetric spaces.

MSC 2000. 41A30, 26A15, 26A16, 26A21, 54E25.
Keywords. Quasi-metric space, semi-Lipschitz function, approximation.

1. PRELIMINARIES

In the last years there have been an increasing interest for the study of quasimetric spaces (spaces with asymmetric metric) motivated by their applications in various branches of mathematics, and especially in computer science. A direction of investigation is to study the possibility to extend to quasi-metric spaces known results in metric spaces (see, for example, [2-[7]).
The following classical result of Baire is well known [8, 9. Every lower semicontinuous real valued function defined on a metric space is the pointwise limit of an increasing sequence of continuous functions. Analyzing the proof of this result (see [9], Th. 2.2-23, p. 84), observe that every element of the increasing sequence is a Lipschitz function. This fact suggest to use the semiLipschitz functions defined in [10, 11, to obtain such a theorem for lower semi-continuous real valued functions defined on a quasi-metric space.
This short Note presents some notions connected with quasi-metric spaces and the result of Baire in this framework.
Let X X XXX be a non-empty set. A function d : X × X → [ 0 , ∞ ) d : X × X → [ 0 , ∞ ) d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty)d:X×X→[0,∞) is called a quasi-metric on X X XXX ([10]) if the following conditions hold:
Q 1 ) d ( x , y ) = d ( y , x ) = 0 Q 1 d ( x , y ) = d ( y , x ) = 0 {:Q_(1))d(x,y)=d(y,x)=0\left.Q_{1}\right) d(x, y)=d(y, x)=0Q1)d(x,y)=d(y,x)=0 iff x = y ; x = y ; x=y;x=y ;x=y;
Q 2 ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) Q 2 d ( x , z ) ≤ d ( x , y ) + d ( y , z ) {:Q_(2))d(x,z) <= d(x,y)+d(y,z)\left.Q_{2}\right) d(x, z) \leq d(x, y)+d(y, z)Q2)d(x,z)≤d(x,y)+d(y,z), for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X.
The function d ¯ : X × X → [ 0 , ∞ ) d ¯ : X × X → [ 0 , ∞ ) bar(d):X xx X rarr[0,oo)\bar{d}: X \times X \rightarrow[0, \infty)d¯:X×X→[0,∞) defined by d ¯ ( x , y ) = d ( y , x ) d ¯ ( x , y ) = d ( y , x ) bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x)d¯(x,y)=d(y,x), for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X is also a quasi-metric on X X XXX, called the conjugate quasi-metric of d d ddd.
The function d s ( x , y ) = max { d ( x , y ) , d ¯ ( x , y ) } d s ( x , y ) = max { d ( x , y ) , d ¯ ( x , y ) } d^(s)(x,y)=max{d(x,y), bar(d)(x,y)}d^{s}(x, y)=\max \{d(x, y), \bar{d}(x, y)\}ds(x,y)=max{d(x,y),d¯(x,y)} is a metric on X X XXX. If d d ddd can take the value + ∞ + ∞ +oo+\infty+∞, then it is called a quasi-distance on X X XXX.
Each quasi-metric d d ddd on X X XXX induces a topology τ ( d ) τ ( d ) tau(d)\tau(d)τ(d) which has as a basis the family of open balls (called forward open balls in [6]):
B + ( x , ε ) := { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . B + ( x , ε ) := { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . B^(+)(x,epsi):={y in X:d(x,y) < epsi},x in X,epsi > 0.B^{+}(x, \varepsilon):=\{y \in X: d(x, y)<\varepsilon\}, x \in X, \varepsilon>0 .B+(x,ε):={y∈X:d(x,y)<ε},x∈X,ε>0.
This topology is called the forward topology of X X XXX ([3, [6]) and is denoted also by τ + τ + tau^(+)\tau^{+}τ+. Observe that the topology τ + τ + tau^(+)\tau^{+}τ+is a T 0 T 0 T_(0)T_{0}T0-topology [10]. If the conditions Q 1 Q 1 Q_(1)Q_{1}Q1 is replaced by Q 1 ′ : d ( x , y ) = 0 Q 1 ′ : d ( x , y ) = 0 Q_(1)^('):d(x,y)=0Q_{1}^{\prime}: d(x, y)=0Q1′:d(x,y)=0 iff x = y x = y x=yx=yx=y, then τ + τ + tau^(+)\tau^{+}τ+is a T 1 T 1 T_(1)T_{1}T1-topology ( 10 , [11]).
The pair ( X , d X , d X,dX, dX,d ) is called a quasi-metric space ( T 0 T 0 T_(0)T_{0}T0-separated, respectively T 1 T 1 T_(1)T_{1}T1-separated). A sequence ( x n ) n ≥ 1 x n n ≥ 1 (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1}(xn)n≥1 in the quasi-metric space ( X , d X , d X,dX, dX,d ) is called τ + τ + tau^(+)\tau^{+}τ+-convergent to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X iff lim n → ∞ d ( x 0 , x n ) = 0 lim n → ∞   d x 0 , x n = 0 lim_(n rarr oo)d(x_(0),x_(n))=0\lim _{n \rightarrow \infty} d\left(x_{0}, x_{n}\right)=0limn→∞d(x0,xn)=0.
Similarly, the topology τ ( d ¯ ) τ ( d ¯ ) tau( bar(d))\tau(\bar{d})τ(d¯) has as a basis the family of open balls:
B − ( x , ε ) : { y ∈ X : d ( y , x ) < ε } , x ∈ X , ε > 0 B − ( x , ε ) : { y ∈ X : d ( y , x ) < ε } , x ∈ X , ε > 0 B^(-)(x,epsi):{y in X:d(y,x) < epsi},x in X,epsi > 0B^{-}(x, \varepsilon):\{y \in X: d(y, x)<\varepsilon\}, x \in X, \varepsilon>0B−(x,ε):{y∈X:d(y,x)<ε},x∈X,ε>0
This topology is denoted also by τ − τ − tau^(-)\tau^{-}τ−.
Definition 1. ([10]). Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space. A function f f fff : X → R X → R X rarrRX \rightarrow \mathbb{R}X→R is called d d ddd-semi-Lipschitz if there exists a number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 (named a d d ddd-semi-Lipschitz constant for f f fff ) such that
(1) f ( x ) − f ( y ) ≤ L d ( x , y ) (1) f ( x ) − f ( y ) ≤ L d ( x , y ) {:(1)f(x)-f(y) <= Ld(x","y):}\begin{equation*} f(x)-f(y) \leq L d(x, y) \tag{1} \end{equation*}(1)f(x)−f(y)≤Ld(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X.
A similar definition can be given for d ¯ d ¯ bar(d)\bar{d}d¯-semi-Lipschitz functions.
Definition 2. The function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R is called ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing if f ( x ) ≤ f ( y ) f ( x ) ≤ f ( y ) f(x) <= f(y)f(x) \leq f(y)f(x)≤f(y), whenever d ( x , y ) = 0 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0.
One denotes by R ≤ d X R ≤ d X R_( <= _(d))^(X)R_{\leq_{d}}^{X}R≤dX the set of all ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing functions on X X XXX. The set R ≤ d X R ≤ d X R_( <= _(d))^(X)\mathbb{R}_{\leq_{d}}^{X}R≤dX is a cone in the linear space R X R X R^(X)\mathbb{R}^{X}RX of real valued functions defined on X X XXX [11].
For a d d ddd-semi-Lipschitz function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R, put
(2) ‖ f | d = sup { ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : d ( x , y ) > 0 , x , y ∈ X } (2) ‖ f d = sup ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : d ( x , y ) > 0 , x , y ∈ X {:(2)||f|_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):d(x,y) > 0,x,y in X}:}\begin{equation*} \|\left. f\right|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: d(x, y)>0, x, y \in X\right\} \tag{2} \end{equation*}(2)‖f|d=sup{(f(x)−f(y))∨0d(x,y):d(x,y)>0,x,y∈X}
Then ‖ f | d ‖ f d ||f|_(d)\|\left. f\right|_{d}‖f|d is the smallest d d ddd-semi-Lipschitz constant of f f fff (see [7], [10, 11]).
Denote also
(3) d -SlipX := { f ∈ R ≤ d X : ‖ f | d < ∞ } (3) d -SlipX  := f ∈ R ≤ d X : ‖ f d < ∞ {:(3)d"-SlipX ":={f inR_( <= d)^(X):||f|_(d) < oo}:}\begin{equation*} d \text {-SlipX }:=\left\{f \in \mathbb{R}_{\leq d}^{X}: \|\left. f\right|_{d}<\infty\right\} \tag{3} \end{equation*}(3)d-SlipX :={f∈R≤dX:‖f|d<∞}
the subcone of the cone R ≤ d X R ≤ d X R_( <= _(d))^(X)\mathbb{R}_{\leq_{d}}^{X}R≤dX, of all d d ddd-semi-Lipschitz real valued functions on ( X , d ) ( X , d ) (X,d)(X, d)(X,d). If θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X is a fixed, but arbitrary element, denote
d − Slip 0 X := { f ∈ d − Slip X : f ( θ ) = 0 } . d − Slip 0 X := { f ∈ d − Slip X : f ( θ ) = 0 } . d-Slip_(0)X:={f in d-SlipX:f(theta)=0}.d-\operatorname{Slip}_{0} \mathrm{X}:=\{f \in d-\operatorname{Slip} \mathrm{X}: f(\theta)=0\} .d−Slip0X:={f∈d−SlipX:f(θ)=0}.
Then the functional ‖ ⋅ | d : d − ‖ ⋅ d : d − ||*|_(d):d-\|\left.\cdot\right|_{d}: d-‖⋅|d:d− Slip 0 X → [ 0 , ∞ ) 0 X → [ 0 , ∞ ) _(0)Xrarr[0,oo)_{0} \mathrm{X} \rightarrow[0, \infty)0X→[0,∞) is an asymmetric norm on d d ddd-Slip 0 X 0 X _(0)X{ }_{0} \mathrm{X}0X, i.e. this functional is subadditive, positively homogeneous and ‖ f | d = ‖ f d = ||f|_(d)=\|\left. f\right|_{d}=‖f|d=
0 iff f = 0 f = 0 f=0f=0f=0. The pair ( d d ddd-Slip 0 X , ‖ ⋅ | d 0 X , ‖ ⋅ d _(0)X,||*|_(d){ }_{0} \mathrm{X}, \|\left.\cdot\right|_{d}0X,‖⋅|d ) is called the "normed cone" of d d ddd-semiLipschitz real valued functions (vanishing at θ θ theta\thetaθ ). The properties of this normed cone are studied in [10, 11.
Definition 3. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space and f : X → R ― f : X → R ¯ f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}}f:X→R―, where R ― = [ − ∞ , + ∞ ] R ¯ = [ − ∞ , + ∞ ] bar(R)=[-oo,+oo]\overline{\mathbb{R}}=[-\infty,+\infty]R―=[−∞,+∞] is equipped with the natural topology. The function f f fff is called τ + τ + tau^(+)\tau^{+}τ+-lower semicontinuous (respectively τ + τ + tau^(+)\tau^{+}τ+-upper semicontinuous) ( τ + τ + tau^(+)\tau^{+}τ+-l.s.c, respectively τ + τ + tau^(+)\tau^{+}τ+-u.s.c., in short) at the point x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, if for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 there exists r > 0 r > 0 r > 0r>0r>0 such that for all x ∈ B + ( x 0 , r ) , f ( x ) > f ( x 0 ) − ε x ∈ B + x 0 , r , f ( x ) > f x 0 − ε x inB^(+)(x_(0),r),f(x) > f(x_(0))-epsix \in B^{+}\left(x_{0}, r\right), f(x)>f\left(x_{0}\right)-\varepsilonx∈B+(x0,r),f(x)>f(x0)−ε (respectively f ( x ) < f ( x 0 ) + ε ) ) f ( x ) < f x 0 + ε {:f(x) < f(x_(0))+epsi))\left.\left.f(x)<f\left(x_{0}\right)+\varepsilon\right)\right)f(x)<f(x0)+ε)).
Similar definitions can be given for τ − τ − tau^(-)\tau^{-}τ−-l.s.c. and τ − τ − tau^(-)\tau^{-}τ−-u.s.c real valued functions on ( X , d ¯ ) ( X , d ¯ ) (X, bar(d))(X, \bar{d})(X,d¯).
Observe that f : X → R ― f : X → R ¯ f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}}f:X→R― is τ + τ + tau^(+)\tau^{+}τ+-l.s.c iff − f − f -f-f−f is τ − τ − tau^(-)\tau^{-}τ−-u.s.c and f f fff is τ + τ + tau^(+)\tau^{+}τ+-u.s.c iff − f − f -f-f−f is τ − τ − tau^(-)\tau^{-}τ−-l.s.c.
The result of Baire in this framework is:
Theorem 4. (Baire). Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space and f : X → R ― f : X → R ¯ f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}}f:X→R― be a τ + τ + tau^(+)\tau^{+}τ+- l.s.c function on X X XXX. Then there exists a sequence ( F n ) n ≥ 1 , F n ∈ d F n n ≥ 1 , F n ∈ d (F_(n))_(n >= 1),F_(n)in d\left(F_{n}\right)_{n \geq 1}, F_{n} \in d(Fn)n≥1,Fn∈d-SlipX such that ( F n ( x ) ) n ≥ 1 , x ∈ X F n ( x ) n ≥ 1 , x ∈ X (F_(n)(x))_(n >= 1),x in X\left(F_{n}(x)\right)_{n \geq 1}, x \in X(Fn(x))n≥1,x∈X, is increasing and lim n → ∞ F n ( x ) = f ( x ) , x ∈ X lim n → ∞   F n ( x ) = f ( x ) , x ∈ X lim_(n rarr oo)F_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} F_{n}(x)=f(x), x \in Xlimn→∞Fn(x)=f(x),x∈X.
Proof. a) Suppose firstly that f ( x ) ≥ 0 f ( x ) ≥ 0 f(x) >= 0f(x) \geq 0f(x)≥0, for all x ∈ X x ∈ X x in Xx \in Xx∈X. For x ∈ X x ∈ X x in Xx \in Xx∈X and n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N, let
(4) F n ( x ) = inf { f ( z ) + n ⋅ d ( x , z ) : z ∈ X } . (4) F n ( x ) = inf { f ( z ) + n ⋅ d ( x , z ) : z ∈ X } . {:(4)F_(n)(x)=i n f{f(z)+n*d(x","z):z in X}.:}\begin{equation*} F_{n}(x)=\inf \{f(z)+n \cdot d(x, z): z \in X\} . \tag{4} \end{equation*}(4)Fn(x)=inf{f(z)+n⋅d(x,z):z∈X}.
Obviously that
(5) 0 ≤ F n ( x ) ≤ f ( x ) + n d ( x , x ) = f ( x ) . (5) 0 ≤ F n ( x ) ≤ f ( x ) + n d ( x , x ) = f ( x ) . {:(5)0 <= F_(n)(x) <= f(x)+nd(x","x)=f(x).:}\begin{equation*} 0 \leq F_{n}(x) \leq f(x)+n d(x, x)=f(x) . \tag{5} \end{equation*}(5)0≤Fn(x)≤f(x)+nd(x,x)=f(x).
If x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X, then
F n ( x ) ≤ f ( z ) + n d ( x , z ) ≤ f ( z ) + n d ( x , y ) + n d ( y , z ) = ( f ( z ) + n d ( y , z ) ) + n d ( x , y ) . F n ( x ) ≤ f ( z ) + n d ( x , z ) ≤ f ( z ) + n d ( x , y ) + n d ( y , z ) = ( f ( z ) + n d ( y , z ) ) + n d ( x , y ) . {:[F_(n)(x) <= f(z)+nd(x","z) <= f(z)+nd(x","y)+nd(y","z)],[=(f(z)+nd(y","z))+nd(x","y).]:}\begin{aligned} F_{n}(x) & \leq f(z)+n d(x, z) \leq f(z)+n d(x, y)+n d(y, z) \\ & =(f(z)+n d(y, z))+n d(x, y) . \end{aligned}Fn(x)≤f(z)+nd(x,z)≤f(z)+nd(x,y)+nd(y,z)=(f(z)+nd(y,z))+nd(x,y).
Taking the infimum with respect to z z zzz, one obtains
F n ( x ) ≤ F n ( y ) + n d ( x , y ) , F n ( x ) ≤ F n ( y ) + n d ( x , y ) , F_(n)(x) <= F_(n)(y)+nd(x,y),F_{n}(x) \leq F_{n}(y)+n d(x, y),Fn(x)≤Fn(y)+nd(x,y),
i.e.
F n ( x ) − F n ( y ) ≤ n ⋅ d ( x , y ) , F n ( x ) − F n ( y ) ≤ n ⋅ d ( x , y ) , F_(n)(x)-F_(n)(y) <= n*d(x,y),F_{n}(x)-F_{n}(y) \leq n \cdot d(x, y),Fn(x)−Fn(y)≤n⋅d(x,y),
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X. This means that ‖ F n | d ≤ n ‖ F n d ≤ n ||F_(n)|_(d) <= n\|\left. F_{n}\right|_{d} \leq n‖Fn|d≤n and F n ∈ d F n ∈ d F_(n)in dF_{n} \in dFn∈d-SlipX, for every n = 1 , 2 , 3 , … n = 1 , 2 , 3 , … n=1,2,3,dotsn=1,2,3, \ldotsn=1,2,3,…
If n ≤ m n ≤ m n <= mn \leq mn≤m, by the definition (4) it follows F n ( x ) ≤ F m ( x ) , x ∈ X F n ( x ) ≤ F m ( x ) , x ∈ X F_(n)(x) <= F_(m)(x),x in XF_{n}(x) \leq F_{m}(x), x \in XFn(x)≤Fm(x),x∈X, so that the sequence ( F n ( x ) ) n ≥ 1 F n ( x ) n ≥ 1 (F_(n)(x))_(n >= 1)\left(F_{n}(x)\right)_{n \geq 1}(Fn(x))n≥1 is increasing and bounded by f ( x ) , x ∈ X f ( x ) , x ∈ X f(x),x in Xf(x), x \in Xf(x),x∈X. Consequently there exists the limit lim n → ∞ F n ( x ) = h ( x ) lim n → ∞   F n ( x ) = h ( x ) lim_(n rarr oo)F_(n)(x)=h(x)\lim _{n \rightarrow \infty} F_{n}(x)=h(x)limn→∞Fn(x)=h(x), and h ( x ) ≤ f ( x ) , x ∈ X h ( x ) ≤ f ( x ) , x ∈ X h(x) <= f(x),x in Xh(x) \leq f(x), x \in Xh(x)≤f(x),x∈X.
In fact h ( x ) = f ( x ) h ( x ) = f ( x ) h(x)=f(x)h(x)=f(x)h(x)=f(x), for every x ∈ X x ∈ X x in Xx \in Xx∈X. Indeed, let n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N and x ∈ X x ∈ X x in Xx \in Xx∈X. By definition (4) of F n ( x ) F n ( x ) F_(n)(x)F_{n}(x)Fn(x), for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, there exists z n ∈ X z n ∈ X z_(n)in Xz_{n} \in Xzn∈X such that
F n ( x ) + ε > f ( z n ) + n d ( x , z n ) ≥ n d ( x , z n ) , F n ( x ) + ε > f z n + n d x , z n ≥ n d x , z n , F_(n)(x)+epsi > f(z_(n))+nd(x,z_(n)) >= nd(x,z_(n)),F_{n}(x)+\varepsilon>f\left(z_{n}\right)+n d\left(x, z_{n}\right) \geq n d\left(x, z_{n}\right),Fn(x)+ε>f(zn)+nd(x,zn)≥nd(x,zn),
and because F n ( x ) + ε ≤ f ( x ) + ε F n ( x ) + ε ≤ f ( x ) + ε F_(n)(x)+epsi <= f(x)+epsiF_{n}(x)+\varepsilon \leq f(x)+\varepsilonFn(x)+ε≤f(x)+ε, it follows
f ( x ) + ε ≥ n d ( x , z n ) f ( x ) + ε ≥ n d x , z n f(x)+epsi >= nd(x,z_(n))f(x)+\varepsilon \geq n d\left(x, z_{n}\right)f(x)+ε≥nd(x,zn)
and then
d ( x , z n ) ≤ 1 n ( f ( x ) + ε ) d x , z n ≤ 1 n ( f ( x ) + ε ) d(x,z_(n)) <= (1)/(n)(f(x)+epsi)d\left(x, z_{n}\right) \leq \frac{1}{n}(f(x)+\varepsilon)d(x,zn)≤1n(f(x)+ε)
For n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞ it follows that the sequence ( z n ) n ≥ 1 z n n ≥ 1 (z_(n))_(n >= 1)\left(z_{n}\right)_{n \geq 1}(zn)n≥1 is τ + τ + tau^(+)\tau^{+}τ+-convergent to x x xxx, and because f f fff is supposed τ + τ + tau^(+)\tau^{+}τ+-l.s.c.,
lim n → ∞ inf f ( z n ) ≥ f ( x ) lim n → ∞   inf f z n ≥ f ( x ) lim_(n rarr oo)i n f f(z_(n)) >= f(x)\lim _{n \rightarrow \infty} \inf f\left(z_{n}\right) \geq f(x)limn→∞inff(zn)≥f(x)
(see [9], p. 127).
Consequently, for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 there exists n 0 ∈ N n 0 ∈ N n_(0)inNn_{0} \in \mathbb{N}n0∈N such that, for every n ≥ n 0 n ≥ n 0 n >= n_(0)n \geq n_{0}n≥n0,
(6) f ( z n ) ≥ f ( x ) − ε . (6) f z n ≥ f ( x ) − ε . {:(6)f(z_(n)) >= f(x)-epsi.:}\begin{equation*} f\left(z_{n}\right) \geq f(x)-\varepsilon . \tag{6} \end{equation*}(6)f(zn)≥f(x)−ε.
By (4) and (6) it follows
F n ( x ) > f ( z n ) − ε + n d ( x , z n ) ≥ f ( z n ) − ε ≥ f ( x ) − 2 ε F n ( x ) > f z n − ε + n d x , z n ≥ f z n − ε ≥ f ( x ) − 2 ε F_(n)(x) > f(z_(n))-epsi+nd(x,z_(n)) >= f(z_(n))-epsi >= f(x)-2epsiF_{n}(x)>f\left(z_{n}\right)-\varepsilon+n d\left(x, z_{n}\right) \geq f\left(z_{n}\right)-\varepsilon \geq f(x)-2 \varepsilonFn(x)>f(zn)−ε+nd(x,zn)≥f(zn)−ε≥f(x)−2ε
for n > n 0 n > n 0 n > n_(0)n>n_{0}n>n0. By (5) one obtains
0 ≤ F n ( x ) ↑ f ( x ) 0 ≤ F n ( x ) ↑ f ( x ) 0 <= F_(n)(x)uarr f(x)0 \leq F_{n}(x) \uparrow f(x)0≤Fn(x)↑f(x)
for all x ∈ X x ∈ X x in Xx \in Xx∈X.
b) Now, let f f fff be a bounded and τ + τ + tau^(+)\tau^{+}τ+-l.s.c function on X X XXX. Then there exists M > 0 M > 0 M > 0M>0M>0 such that | f ( x ) | ≤ M | f ( x ) | ≤ M |f(x)| <= M|f(x)| \leq M|f(x)|≤M, for all x ∈ X x ∈ X x in Xx \in Xx∈X. Denoting g ( x ) = f ( x ) + M , x ∈ X g ( x ) = f ( x ) + M , x ∈ X g(x)=f(x)+M,x in Xg(x)=f(x)+M, x \in Xg(x)=f(x)+M,x∈X, one obtains g ( x ) ≥ 0 g ( x ) ≥ 0 g(x) >= 0g(x) \geq 0g(x)≥0, and g g ggg is τ + τ + tau^(+)\tau^{+}τ+-l.s.c., on X X XXX.
From the part a), there exists the sequence ( G n ) n ≥ 1 , G n ∈ d G n n ≥ 1 , G n ∈ d (G_(n))_(n >= 1),G_(n)in d\left(G_{n}\right)_{n \geq 1}, G_{n} \in d(Gn)n≥1,Gn∈d-SlipX, n = 1 , 2 , 3 , … n = 1 , 2 , 3 , … n=1,2,3,dotsn= 1,2,3, \ldotsn=1,2,3,… such that 0 ≤ G n ( x ) ↑ g ( x ) = f ( x ) + M 0 ≤ G n ( x ) ↑ g ( x ) = f ( x ) + M 0 <= G_(n)(x)uarr g(x)=f(x)+M0 \leq G_{n}(x) \uparrow g(x)=f(x)+M0≤Gn(x)↑g(x)=f(x)+M. This means that the sequence ( F n ) n ≥ 1 , F n = G n − M , n = 1 , 2 , 3 … F n n ≥ 1 , F n = G n − M , n = 1 , 2 , 3 … (F_(n))_(n >= 1),F_(n)=G_(n)-M,n=1,2,3dots\left(F_{n}\right)_{n \geq 1}, F_{n}=G_{n}-M, n=1,2,3 \ldots(Fn)n≥1,Fn=Gn−M,n=1,2,3… is increasing and converges in every point x x xxx of X X XXX to f ( x ) f ( x ) f(x)f(x)f(x). Moreover | F n ( x ) | ≤ M , x ∈ X , n = 1 , 2 , 3 , … F n ( x ) ≤ M , x ∈ X , n = 1 , 2 , 3 , … |F_(n)(x)| <= M,x in X,n=1,2,3,dots\left|F_{n}(x)\right| \leq M, x \in X, n=1,2,3, \ldots|Fn(x)|≤M,x∈X,n=1,2,3,….
c) Now, we consider the general case, i.e., f : X → R ― f : X → R ¯ f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}}f:X→R― is an arbitrary τ + τ + tau^(+)\tau^{+}τ+-l.s.c. function.
Using the Baire function (see [1]) φ : [ − ∞ , + ∞ ] → [ − 1 , 1 ] φ : [ − ∞ , + ∞ ] → [ − 1 , 1 ] varphi:[-oo,+oo]rarr[-1,1]\varphi:[-\infty,+\infty] \rightarrow[-1,1]φ:[−∞,+∞]→[−1,1],
φ ( x ) = { − 1 , for x = − ∞ x 1 + | x | , for − ∞ < x < ∞ 1 , for x = + ∞ φ ( x ) = − 1 ,  for  x = − ∞ x 1 + | x | ,  for  − ∞ < x < ∞ 1 ,  for  x = + ∞ varphi(x)={[-1","" for "x=-oo],[(x)/(1+|x|)","" for "-oo < x < oo],[1","" for "x=+oo]:}\varphi(x)=\left\{\begin{array}{l} -1, \text { for } x=-\infty \\ \frac{x}{1+|x|}, \text { for }-\infty<x<\infty \\ 1, \text { for } x=+\infty \end{array}\right.φ(x)={−1, for x=−∞x1+|x|, for −∞<x<∞1, for x=+∞
which is a Lipschitz increasing isomorphism, it follows that φ ∘ f : R ― → [ − 1 , 1 ] φ ∘ f : R ¯ → [ − 1 , 1 ] varphi@f: bar(R)rarr[-1,1]\varphi \circ f: \overline{\mathbb{R}} \rightarrow[-1,1]φ∘f:R―→[−1,1] is bounded and τ + − τ + − tau^(+)-\tau^{+}-τ+−l.s.c. on X X XXX.
By the previous point b), there exists a sequence ( H n ) n ≥ 1 H n n ≥ 1 (H_(n))_(n >= 1)\left(H_{n}\right)_{n \geq 1}(Hn)n≥1 with H n ∈ d H n ∈ d H_(n)in dH_{n} \in dHn∈d-SlipX such that
H n ( x ) ↑ ( φ ∘ f ) ( x ) , x ∈ X . H n ( x ) ↑ ( φ ∘ f ) ( x ) , x ∈ X . H_(n)(x)uarr(varphi@f)(x),x in X.H_{n}(x) \uparrow(\varphi \circ f)(x), x \in X .Hn(x)↑(φ∘f)(x),x∈X.
Consequently, the sequence ( F n ) n ≥ 1 , F n ( x ) = ( φ − 1 ∘ H n ) ( x ) , x ∈ X F n n ≥ 1 , F n ( x ) = φ − 1 ∘ H n ( x ) , x ∈ X (F_(n))_(n >= 1),F_(n)(x)=(varphi^(-1)@H_(n))(x),x in X\left(F_{n}\right)_{n \geq 1}, F_{n}(x)=\left(\varphi^{-1} \circ H_{n}\right)(x), x \in X(Fn)n≥1,Fn(x)=(φ−1∘Hn)(x),x∈X, is increasing and lim n → ∞ F n ( x ) = f ( x ) , x ∈ X lim n → ∞   F n ( x ) = f ( x ) , x ∈ X lim_(n rarr oo)F_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} F_{n}(x)=f(x), x \in Xlimn→∞Fn(x)=f(x),x∈X.
For τ + τ + tau^(+)\tau^{+}τ+-u.s.c functions on X X XXX, one obtains:
Theorem 5. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space and f : X → R ― f : X → R ¯ f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}}f:X→R― a τ + τ + tau^(+)\tau^{+}τ+-u.s.c. function. Then there exists a sequence ( G n ) n ≥ 1 , G n ∈ d G n n ≥ 1 , G n ∈ d (G_(n))_(n >= 1),G_(n)in d\left(G_{n}\right)_{n \geq 1}, G_{n} \in d(Gn)n≥1,Gn∈d-SlipX, n = 1 , 2 , 3 , … n = 1 , 2 , 3 , … n=1,2,3,dotsn=1,2,3, \ldotsn=1,2,3,… such that ( G n ( x ) ) n ≥ 1 , x ∈ X G n ( x ) n ≥ 1 , x ∈ X (G_(n)(x))_(n >= 1),x in X\left(G_{n}(x)\right)_{n \geq 1}, x \in X(Gn(x))n≥1,x∈X, is monotonically decreasing and lim n → ∞ G n ( x ) = f ( x ) , x ∈ X lim n → ∞   G n ( x ) = f ( x ) , x ∈ X lim_(n rarr oo)G_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} G_{n}(x)= f(x), x \in Xlimn→∞Gn(x)=f(x),x∈X.
Proof. If f f fff is τ + τ + tau^(+)\tau^{+}τ+-u.s.c., then − f − f -f-f−f is τ − τ − tau^(-)\tau^{-}τ−-l.s.c on X X XXX. By Theorem 1, there exists a sequence ( F n ) n ≥ 1 F n n ≥ 1 (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geq 1}(Fn)n≥1 in d ¯ d ¯ bar(d)\bar{d}d¯-SlipX, monotonically increasing and pointwise convergent to − f − f -f-f−f. Then the sequence G n = − F n , n = 1 , 2 , … G n = − F n , n = 1 , 2 , … G_(n)=-F_(n),n=1,2,dotsG_{n}=-F_{n}, n=1,2, \ldotsGn=−Fn,n=1,2,…, has the following properties: G n = − F n ∈ d G n = − F n ∈ d G_(n)=-F_(n)in dG_{n}=-F_{n} \in dGn=−Fn∈d-SlipX, and G n ( x ) ↓ f ( x ) , lim n → ∞ G n = f ( x ) , x ∈ X G n ( x ) ↓ f ( x ) , lim n → ∞   G n = f ( x ) , x ∈ X G_(n)(x)darr f(x),lim_(n rarr oo)G_(n)=f(x),x in XG_{n}(x) \downarrow f(x), \lim _{n \rightarrow \infty} G_{n}=f(x), x \in XGn(x)↓f(x),limn→∞Gn=f(x),x∈X.

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Received by the editors: March 27, 2008.
  1. *This work was supported by MEdC under grant 2CEEX06-11-96/19.09.2006.
    † † ^(†){ }^{\dagger}† "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro, cmustata2001@yahoo.com.
2008

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