Erratum : A new setting for potential theory (part 1)

A NNALES DE L ’ ^INSTITUT F ^OURIER

K. L. C ^HUNG K. M ^URALI R ^AO

Erratum : A new setting for potential theory (part 1)

Annales de l’institut Fourier, tome 30, n

3 (1980), p. 1-2 (feuilles volantes)

<

>

© Annales de l’institut Fourier, 1980, tous droits réservés.

L’accès aux archives de la revue « Annales de l’institut Fourier » (

) implique l’accord avec les conditions gé- nérales d’utilisation (

). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

university scientifique et medicate de grenoble

annales de Hnstitut fourier

E R R A T U M

"A NEW SETTING FOR POTENTIAL THEORY (part 1)"

Article paru dans Ie tome 30 (1980), fascicule 3, pp. 167-198

M6moire de K. L. CHUNG & K. MURALI RAO

On p. 181, (26) should read as follows :

<20) ^K = ^ •

Proof that (a) =» (b) should be revised as follows.

Suppose Z is polar and K be given. Let L be compact. Lc K 0 Z0 . By Proposition 1 of §1. there exists h such that h > 0 everywhere and Uh -s 1 . Let s - P..Uh . then s = lim I Pp. Uh where D U K Bv

K n "n n "

Corollaries 1 and 4 of Theorem 2 (continued), we have V - U ^ . I^CD^ . ^ ( Z ) = 0 .

Hence s = li^n V\s^ , and U|j^ ^ P^ Uh < oo for all n . Apply Theorem 2 (continued) under (c ) to obtain (n^ ] converging vaguely to (JL . such that s - UH and p(Z) = 0 . the last assertion by (6) of Theorem 2.

We have ^JL c K by vague convergence. Thus

s = up . i? c K . n(Z) = o .

and therefore s ^ WH . For any (open) G 3 K . we have then P^s = P^WH = Wn = s

where the second equation is due to the round property of w and the fact H is supported by Kc G . Thus by the argument on p. 70 of (51 :

.../...

- 2

I V^ j

0 ' y111 - W^ ' E < L ^t^( 1^ 1 ; ^ ^K ; ^G^ KVKr

which implies that

vx : P^T -T i X f r ^ e K Y K ^ = o .

This implies easily that for any f ^ bC :

W '- Y

which is (26).

N . B . The mistake was to suppose that P.^P.,1 = P,,l implies P P. = P . tx K K G K K This was partly caused by a statement on p. 71 of (5l which apparently asserts that P/,P.. ^ P., in general. Dellacherie gave a trivial counter-

0 1\ Iv

example to the last assertion, which is left as an exercise.

First display on p. 168 should read : lim P ^ T - o O . <oo) = 0 . t-oo K t