Reference for proof that \$C_b^* = rba\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:36:04Z http://mathoverflow.net/feeds/question/70611 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70611/reference-for-proof-that-c-b-rba Reference for proof that \$C_b^* = rba\$ Mark Peletier 2011-07-18T11:34:51Z 2011-08-11T14:31:33Z <p>The following theorem seems to have folk status:</p> <p>The topological dual of the space \$C_b(X)\$ of bounded continuous functions on a topological space \$X\$ is isomorphic to the space \$rba(X)\$ of finite, regular, finitely additive Borel set functions.</p> <p>This fact is often mentioned (for instance in the answer to <a href="http://mathoverflow.net/questions/44183/dual-of-bounded-uniformly-continuous-functions" rel="nofollow">http://mathoverflow.net/questions/44183/dual-of-bounded-uniformly-continuous-functions</a>) but I'm having great difficulty actually finding a reference. Often Dunford &amp; Schwartz is mentioned as a reference; D&amp;S defines \$rba\$, but doesn't prove the connection to the dual of \$C_b\$. <a href="http://www.jstor.org/pss/1989829" rel="nofollow">Hildebrandt 1934</a> proves a characterization in terms of limits of Stieltjes integrals, but that is still some steps away from the characterization above. I haven't been able to find anything coming closer than this.</p> <p>Does anyone know of a real proof of this statement? Am I maybe overlooking a very simple proof?</p> http://mathoverflow.net/questions/70611/reference-for-proof-that-c-b-rba/72618#72618 Answer by dan232 for Reference for proof that \$C_b^* = rba\$ dan232 2011-08-10T20:11:16Z 2011-08-10T20:11:16Z <p>In the answer you mentioned, the space \$X\$ is metrizable, hence normal, so the proof from Dunford &amp; Schwartz that appeared in the comments is aplicable in that case.</p> http://mathoverflow.net/questions/70611/reference-for-proof-that-c-b-rba/72662#72662 Answer by Ljubomir Cukic for Reference for proof that \$C_b^* = rba\$ Ljubomir Cukic 2011-08-11T07:53:38Z 2011-08-11T07:53:38Z <p>The topological dual of the space of bounded continuous functions on a topological space X is isomorphic to the space of finite, zero set regular, finitely additive <em>Baire</em> set functions; see: R. F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 2 (1983), 97–190 (a proof is on pp. 115-117).</p>