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For integrable random variable $X$ and nested sub-sigma-algebras $\mathcal{H}\subseteq \mathcal{G}$, the tower property saysThe finer-information conditional mean can be averaged within each coarser group, recovering the coarser-information conditional mean. Conversely, conditioning an already $\mathcal{H}$-measurable conditional mean on $\mathcal{G}$ leaves it unchanged. Both orders therefore recover $\mathbb{E}[X\mid \mathcal{H}]$ when the information is nested.^[ref]

For the finite example in law of total expectation, exact knowledge of $X$ gives the finer mean $X$ itself. Averaging within the groups $A,B$ recovers two and eight. Forgetting the group then gives $13/2$. The nesting condition is essential; iterated conditional expectation over nonnested sigma-algebras describes why arbitrary successive conditioning cannot simply be replaced by conditioning on an intersection.

The equation fails for general nonnested sigma-algebras. On the four-point space $\Omega =\{1,2,3,4\}$ with uniform probability, letand take $X=1_A$. The intersection $\mathcal{G}\cap \mathcal{H}$ is trivial, soBut $X$ is $\mathcal{G}$-measurable andwhich is zero on $B^c$ and is not almost surely $1/2$. This exhibits the failure of iterated conditional expectation over nonnested sigma-algebras.