Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 261–262

A REMARK ON THE UNIQUENESS OF QUASI CONTINUOUS FUNCTIONS

Tero Kilpeläinen

University of Jyväskylä, Department of Mathematics P.O. Box 35, FIN-40351 Jyväskylä, Finland; terok@math.jyu.fi

Abstract. We give a simple argument showing that two quasi continuous functions that agree almost everywhere coincide in fact quasi everywhere.

1. Introduction

By a capacity on a topological space X we understand a countably subadditive, monotone set function 𝒞 defined on the subsets of X with values on the interval [0, ∞] such that 𝒞 (∅) = 0. Moreover, a capacity 𝒞 is called an outer capacity if

𝒞 (A) = inf{𝒞 (G) : G is open and A ⊂ G}.

Furthermore a function f: X → \overline{R} is termed 𝒞 -quasi continuous if for each ε > 0 there is a set G ⊂ X such that 𝒞 (G) < ε and the restriction to X \ G of f is continuous. Evidently, G can be chosen to be open if 𝒞 is outer. See [F] for properties of abstract capacities and quasi topologies.

An example of an outer capacity is the p-capacity cap_{p} on R^{n}:

cap_{p}(E) = inf ∫_{R^{n}}(|∇φ|^{p} + |φ|^{p}) dx,

where the infimum is taken over all Sobolev functions φ ∈ W^{1,p}(R^{n}) such that φ ≥ 1 a.e. on an open neighborhood of E. In that case it is well known (see [AH], [DL], [HKM], [MK]) that cap_{p} -quasi continuous functions enjoy the following uniqueness property: Let f and g be cap_{p} -quasi continuous functions on an open set Ω ⊂ R^{n} such that f = g almost everywhere on Ω (w.r.t. Lebesgue measure). Then f = g cap_{p} -quasi everywhere on Ω. These proofs are based on fine properties of capacitary potentials. Recall that a property is said to hold 𝒞 -quasi everywhere if it holds except on a set E with 𝒞 (E) = 0. Our purpose in this note is to give an elementary and short proof in the abstract setting:

1991 Mathematics Subject Classification: Primary 31C15, 31C45, 31D05, 46E39.

The research is financed by the Academy of Finland.

Theorem. Suppose that 𝒞 is an outer capacity on X and μ is a nonnegative, monotone set function on X such that the following compatibility condition is safisfied: if G is open and μ(E) = 0, then

𝒞 (G) = 𝒞 (G \ E).

Let f and g be 𝒞 -quasi continuous functions on X such that

μ({x : f(x) ≠ g(x)}) = 0.

Then f = g 𝒞 -quasi everywhere on X.

Remarks. It is clear from the definition that the usual capacities associated with Sobolev spaces, the p-capacity cap_{p} in particular, and the Lebesgue measure satisfy the compatibility condition.

Moreover, the proof does not make use of the subadditivity of the capacity. Proof. Fix ε > 0 and choose an open set G such that 𝒞 (G) < ε and that

the restrictions to X \ G of f and g are continuous. Thus, by topology, there is an open subset U of X with

U \ G = {x ∉ G : f(x) ≠ g(x)}.

Since both U and G are open and since μ(U \G) = 0, the compatibility condition implies that 𝒞 (G ∪ U) = 𝒞 (G) < ε. The theorem follows since

{x : f(x) ≠ g(x)} ⊂ G ∪ {x ∉ G : f(x) ≠ g(x)} = G ∪ U.

References

[AH] Adams, D.R., and L.I. Hedberg: Function Spaces and Potential Theory. - Springer-Verlag, Berlin, 1996.

[DL] Deny, J., and J.L. Lions: Les espaces du type de Beppo Levi. - Ann. Inst. Fourier 5, 1953–54, 305–370.

[F] Fuglede, B.: The quasi topology associated with a countable subadditive set function. - Ann. Inst. Fourier 21:1, 1971, 123–169.

[HKM] Heinonen, J., T. Kilpeläinen, and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Oxford University Press, Oxford, 1993.

[MK] Maz’ya, V.G., and V.P. Khavin: Non-linear potential theory. - Russian Math. Surveys 27, 1972, 71–148.

Received 31 January 1997