We can actually extend what we’ve been doing with Hölder’s inequality and Minkowski’s inequality a little further. Given a metric space 
, we’ve already discussed the idea of an “essentially bounded” function — one for which there is some real constant 
so that 
for almost all 
. We will write 
for the collection of essentially bounded functions on the measure space. It should be clear that these form a vector space.
We also discussed the “essential supremum” 
of an essentially bounded function. We’ll now write this as 
, suggesting that it’s a norm. And it’s clear that 
, and that 
if and only if 
almost everywhere. Verifying the triangle identity is exactly Minkowski’s inequality.
And, indeed, we know that 
and 
a.e., so 
a.e., so whatever the least such essential upper bound is smaller still. That is, 
.
Now for Hölder’s inequality. For this purpose we consider 
, and thus 
, which means that 
and 
are Hölder-conjugates. Thus our assertion is that if 
is integrable and 
is essentially bounded, then 
is integrable and 
. Indeed, we know that , and so 
— both inequalities holding almost everywhere. From this, we conclude that
as we asserted. From now on, we’ll allow 
(and 
) whenever we’re talking about a Hölder-conjugate pair or 
-space.
August 30, 2010
Posted by John Armstrong |
Analysis, Measure Theory |
1 Comment
