Does any one know an example of a bilinear functional $B:C(X)\times C(Y)\to {\bf R}$ ($X$ and $Y$ are open subsets of Euclid spaces) which cannot be extended continuously to a measure $\mu:C(X\times Y)\to {\bf R}$ (in the sense that $<\mu, f \times g >=B(f,g)$ where $f\in C(X)$, $g\in C(Y)$).

Many thanks!

  • 1
    $\begingroup$ To my eyes, your question would seem more natural if $X$ and $Y$ were compact (measures correspond to functionals on $C_0$ or $C_c$). In any case, I think the word to look for online or in MathSciNet is "bimeasure" $\endgroup$ – Yemon Choi Jul 1 '14 at 15:14
  • $\begingroup$ We probably also want to assume that $B$ is bounded? $\endgroup$ – Christian Remling Jul 2 '14 at 0:05

Take $x_1,\ldots,x_n \in X$ and $y_1,\ldots,y_n\in Y$ and consider the bilinear functional $$B_n(f,g)=\sum_{i\neq j}(i-j)^{-1}f(i)g(j).$$ By Hilbert's inequality, $|B_n(f,g)|\le\pi\|f\|_2\|g\|_2\le\pi n \|f\|_\infty\|g\|_\infty$. On the other hand, $$\sum_{i \neq j}|i-j|^{-1}=2\sum_{k=1}^{n-1}\frac{n-k}{k}\sim2n\log n.$$ So the measure $\mu_n$ representing $B_n$ has norm of order $n\log n$. Hence, for $\lambda_n>0$ such that $\sum_n \lambda_n n <\infty$ and $\sum_n \lambda_n n\log n=\infty$, one obtains a bounded bilinear form $\sum_n \lambda_n B_n$ which does not extends to a bounded measure. This construction is somewhat canonical. If you want to study it more, look up Grothendieck's inequality.

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