the tower (smoothing) property

Imagine you estimate something in two stages: first you get a coarse summary of the data, then later a finer, more detailed view. The tower property says that if you average your detailed estimate back down to the coarse level, you get exactly the coarse estimate you would have made directly. Refining and then re-coarsening lands you where the coarse view alone would have. Information you later threw away cannot have mattered to the coarse average.

Formally, when one sigma-algebra H sits inside a larger one G (H is the coarser information, G the finer), E[ E[X given G] given H ] = E[X given H]. The smaller (coarser) sigma-algebra wins: the outer, coarser conditioning swallows the inner, finer one. The two extreme corollaries are the ones you use daily. Take H to be the trivial sigma-algebra: E[ E[X given G] ] = E[X] — averaging the conditional expectation recovers the plain mean, which is exactly the law of total expectation. Conditioning on a single variable, this reads E[X] = E[ E[X given Y] ], the workhorse of nearly every clever expectation computation: condition on something convenient, average inside, then average out.

The geometric reason is one line: in the L^2 picture, projecting onto the big subspace and then onto the small one is the same as projecting straight onto the small one — the first projection is wasted because the second discards the very detail it added. This is why the property is also called 'smoothing'. The direction matters: the identity only collapses when the OUTER conditioning is the coarser one (H inside G). Reverse the order and E[ E[X given H] given G ] simply equals E[X given H] again (since E[X given H] is already G-measurable), which is a different and weaker statement.