This post talks about the justification of convolution from measure theory.
Measure Theory
We first introduce the following three theorems.
Theorem 1:
Suppose is a measurable function on
. Then the function
is measurable on
.
Theorem 2:
If is a measurable function on
, then the function
is measurable on
.
The theorems above ensure the first step of the justification later. Moreover, one classic version of the well-known Fubini’s Theorem will be stated.
Fubini’s Theorem:
We first rewrite
Suppose is a non-negative measurable function on
. Then for almost every
-
- The slice
is measurable on
- The function defined by the following is measurable on
- Moreover, we have
- The slice
Now we will approach the topic by the following steps. Suppose that and
are measurable functions on
.
Step 1:
is measurable on
.
Proof:
By theorem 2, if is measurable on
, then
is measurable on
. Also, by theorem 1, if
is measurable on
, then
is measurable on
. Therefore,
is the product of two measurable functions, and it is indeed measurable.
Step 2:
If and
are integrable on
, then
is integrable on
.
Proof:
From step 1, the function is measurable, then so is
. By Fubini’s Theorem,
and the function is integrable.
Step 3:
Define the convolution of and
to be
Then is well defined for almost every
. That is,
is integrable on
for almost every
.
Proof:
Since from step 2, is integrable on
, then by Fubini’s theorem,
is integrable on
for any fixed
, i.e., almost every
.
As a matter of fact, various conditions on and
can guarantee that
is well defined for almost everywhere. For instance, one other than above is that if
is bounded and compactly supported,
can be any locally integrable function. To see how this concept can be approached from different aspects, check out the other two volumes of this series (this post and this post).
Most of the content above is based on Real Analysis by Elias M. Stein abd Rami Shakarchi and Real Analysis by Gerald B. Folland. I also learned a lot from the class MAT206 taught by Professor Adam Jacob. Thanks for his kindly teaching!
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