Inner Measures
A quick one for today.
In analogy with the outer measure 
induced on the hereditary 
-ring 
by the measure 
on the -ring 
, we now define the “inner measure” 
that induces on the same hereditary -ring . We’ve seen that the outer measure is
Accordingly, the inner measure is a set function defined by
In a way, the properties of are “dual” to those of . The easy ones are the same: it’s non-negative, monotone, and 
.
We could also define in terms of the completed measure. Since 
, it’s clear
On the other hand, the definition of the completion says that for every 
there is a 
with 
and 
, and so this is actually an equality.
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April 8, 2010 -
Posted by John Armstrong |
Analysis, Measure Theory
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[...] a measure space , we will routinely use without comment the associated outer measure and inner measure on the hereditary -ring [...]
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