Quick description
From Fubini's theorem one has the identity
for non-negative . Thus, to estimate an integral, one way to do this is to control the measure of the level sets
for different values of
.
A slight variant: the norm of a function
can be expressed by the formula
As another variant, one can view (1) as a means to decompose any non-negative function as a superposition of indicator functions:

Prerequisites
Measure theory
Example 1
Let be a finite set, and let
be a function. Show that

for any .
Solution: we may as well normalize , thus

If we insert this bound directly into (2) (using counting measure instead of
) we obtain a logarithmic divergence. But we can improve the bound in two ways. Firstly, when
, then
, since there is no other non-negative integer less than
. Secondly, we also have the trivial upper bound of
, which is superior when
. If we then use "divide and conquer" and partition the integral on the right-hand side of (2) into the regions
,
, and
, one obtains the claim.
Example 2
Show that if a function lies in the weak
spaces
and
for some measure space
and some exponents
, then it lies in the strong
spaces
for all
.
We need to use "divide and conquer" efficiently in order to 'interpolate' the information we have at the endpoint weak spaces. The idea is that when the function
is large, say
, then the
norms increase when
increases which is the same as saying that
whenever
and
. This indicates that we should split the integral
into two parts; one integral over the set where
is large and one over its complement. Then the integral over the region where
is large is controlled by the weak
norm of
and the integral where
is small is controlled by the weak
norm of
.
This can be done in a very elegant fashion by using the description (2) of the norm of a function. Indeed we can write

For the first term (which corresponds to the set where is small) we use the weak
estimate

Similarly we get for the second term

Thus we have

and in particular
More efficiently, one could split the integral at some point instead of the point
and then optimize in the parameter
.
General discussion
This method is closely related to double counting and "interchange integrals or sums".
The method also combines well with dyadic decomposition. Indeed, one easily verifies that is comparable to
, and more generally
is comparable (up to constants depending on
) to
.
The Marcinkiewicz interpolation theorem relies heavily on these sorts of level set decompositions.