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meaning of additivity when defining signed measure
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HopfZ
science forum beginner


Joined: 30 Jun 2006
Posts: 4

PostPosted: Thu Jul 06, 2006 12:43 pm    Post subject: Re: meaning of additivity when defining signed measure Reply with quote

HopfZ wrote:
Quote:
A signed measure on a measurable space (X,B) in Royden's Real Analysis
(third edition) is defined as a function v:2^B -> [-oo,+oo] satisfying

a typo: 2^B should be replaced with B.

Quote:
the following three conditions:
1. v assumes at most one of the values +oo, -oo (so to avoid oo - oo.
oo means infinity)
2. v(\emptyset) = 0
3. v( \union_i E_i ) = \sum_i v(E_i) for any sequence E_i of disjoint
measurable sets, "the equality is taken to mean that the series on the
right converges absolutely if v(\union E_i) is finite and that it
properly diverges otherwise."

My question is what is the precise meaning of the phrase "properly
diverges otherwise" in the third condition.
I suspect the meaning is that if the left side is +oo, the partial sums
on the right approaches +oo. Similarly for -oo case.
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HopfZ
science forum beginner


Joined: 30 Jun 2006
Posts: 4

PostPosted: Thu Jul 06, 2006 12:36 pm    Post subject: meaning of additivity when defining signed measure Reply with quote

A signed measure on a measurable space (X,B) in Royden's Real Analysis
(third edition) is defined as a function v:2^B -> [-oo,+oo] satisfying
the following three conditions:
1. v assumes at most one of the values +oo, -oo (so to avoid oo - oo.
oo means infinity)
2. v(\emptyset) = 0
3. v( \union_i E_i ) = \sum_i v(E_i) for any sequence E_i of disjoint
measurable sets, "the equality is taken to mean that the series on the
right converges absolutely if v(\union E_i) is finite and that it
properly diverges otherwise."

My question is what is the precise meaning of the phrase "properly
diverges otherwise" in the third condition.
I suspect the meaning is that if the left side is +oo, the partial sums
on the right approaches +oo. Similarly for -oo case.
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