Let a banach space. I want to show the following: For any borel measure , the map given by is measurable.

I believe this can be done in a number of ways, one of which is proving that the set for any is contained in (borel sigma-algebra of )

I think proving this, can not be done without using the fact that is induced by some norm . If we take for example with Euclidian metric, ( lebesgue measure) it is obvious is contained in as or for any . But for any random metric , this is not obvious...

A norm-induced metric has (as I recall) 2 extra properties. 1. translation invariance and 2.

I hope what I think is correct here...so suppose for some then for any we have as well...

It's just a hunch, i have no clue actually...as I don't know what properties might have. (except for the standard properties)

Any idea what I'm missing here?

Thank you

(good to see u guys back online though...btw: can it be implemented that $ x^2 $ is valid latex-code here, instead of wraps..it's sometimes

quite bothersome to write tags everywhere :P)