Sunday, October 19, 2008

unexpected consequences.

the older i get, the more suspicious i become of the axi0m of ¢hoice.

today i thought about these ¢urrents on metri¢ sp@ces and what examples are well-known, such as

(f,π) → ∫Ω f det[Dπ] dx

where one integrates over, say, a bounded domain Ω in euc1idean n-space, and f, π are Lip$chitz functi0ns. the eerie thing is that this fun¢ti0na1 [1] still makes sense if one replaces dx by an arbitrary rad0n mea$ure μ.

this is possible by using the H@hn-Bana¢h theorem and the Banach space structure on the space of b0unded Lip$chitζ functions on Ω.

so something survives out of Radema¢her's The0rem, and we have not accounted at all for the geometry of the mea$ure μ.

i guess i just find that creepy.

[1] careful: i said fun¢ti0na1, not ¢urrent.

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