## Fubini-Tonelli principle

Fubini-Tonelli principle: Imagine that you have a saw-shaped “square” with very deep sawteeth and small area. If we integrate iteratedly, then in one order, in the inner integral we can only bound with $1$ and thus the estimate would be $1$, which is very rough; in the other order, in the sparcity can be easily detected and hence we get a more accurate estimate. This corresponds to Tonelli principle which guides us to first detect the sparsity. Similarly, cancellation may be easily detected from one perspective, but not from another. In this case we should integrate first the cancellation coordiate. And this corresponds to Fubini principle.

Integration by parts: is in some sense also a Fubini principle. Take summation by part for example, if we interprete the summation as total area $A$ of consecutive (on $x$ axis) growing rectangles, then summation by parts tells us that the area is the area of the eveloping rectangle minus the sum area $B'$ of the complementary consecutive (on $y$ axis) growing rectangles. If we assume that the area of the eveloping rectangle vanishes (which is usually the case), then it tells us the $|B|=|B'|$. However, it is usually easier to get a more precise estimate for $B'$ since we are detecting the cancellation from an other perspective.

Integrate along cancellation/sparsity: Many tricks, such as polar coordinate, integration through upper level set (whose importance is well known), co-area formula, etc. All of these can be seen as Fubini-Tonelli principle. Differentiate along cancellation: In a broader sense, the principle can even be applied to PDEs. To read certain PDEs, people use the characteristics along which significant cancellation happens so that one can reduce the PDE into ODE, or reduce the order of PDE, say two, into first order stochastic ODE.

Randomization trick: Sometimes we know cancellation/sparsity appears somewhere but they are hard to identify. Then one trick to tackle the problem is to let a random vector detect the cancellation/sparsity for us. We can employ suitable randamization to obtain an extra “axis”.  Integrating first along the random axis (i.e. taking the average),  the cancellation/sparsity might be detected. Example includes random series, spherical projection, random (dyadic) grid, etc.