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Outside of the zero measure and counting measure, what are some examples of of measures over the space $L^2(\Omega)$ where $\Omega$ is some subset of $\mathbb{R}^n$? Mostly, I'm looking for concrete examples of measures in infinite dimensional space outside of stochastic measures like the Gaussian measure. Certainly, as this question answered:

Measure on Hilbert Space

there's not a Lebesgue-like measure. I'm really just looking for a combination of both simple and moderately interesting examples.

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wyer33
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  • The Hilbert cube ${x\in \ell^2\colon |x_i|\le 1/i}$ supports a product measure (product of linear measures on each edge of the cube). You can move this into $L^2$ by Fourier transform to produce something exotic-looking. –  Jun 05 '12 at 15:39

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