Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 26I (Probability and Measure)

A lambda-system contains the whole space and is closed under complements and countable disjoint unions. Let $\mathcal{D}$ be the smallest lambda-system containing a pi-system $\mathcal{P}$. For $A\in \mathcal{P}$, the family of $B\in \mathcal{D}$ with $A\cap B\in \mathcal{D}$ is a lambda-system containing $\mathcal{P}$, hence is all of $\mathcal{D}$. Fixing $B\in \mathcal{D}$ and repeating the argument shows $\mathcal{D}$ is closed under intersections. A lambda-system closed under intersections is a sigma-algebra, so $\mathcal{D}=\sigma (\mathcal{P})$. This proves the pi-lambda theorem. If two probability measures agree on $\mathcal{P}$, the sets on which they agree form a lambda-system; thus they agree on $\sigma (\mathcal{P})$.

The nonnegative form of Fubini's theorem, also called Tonelli theorem, states that for a nonnegative product-measurable function on two sigma-finite measure spaces, the product integral equals either iterated integral, allowing the value infinity.

For Lebesgue measure on the plane, a bounded interval rectangle $I\times J$ has measure $|I||J|$. Its preimage under $f$ is $(\lambda I)\times ({\lambda }^{-1}J)$, which has the same measure. Under $g$, its vertical slice at $x\in I$ is $J-sx$, whose one-dimensional Lebesgue measure is $|J|$ by translation invariance. Tonelli therefore gives $\mu (g^{-1}(I\times J))=|I||J|$. Horizontal slices likewise prove the equality for $h$. The interval rectangles form a generating pi-system. Sigma-finite measure uniqueness, obtained by the finite-measure pi-lambda argument on an increasing rectangular exhaustion, extends these equalities to all Borel sets. Completion extends them to all Lebesgue-measurable sets, since Borel null sets and their subsets remain null.

Writing ${\lambda }^2=c$, matrix multiplication givesThus rotations through angles in $[0,\pi /2)$ preserve measure. Their inverses and compositions do too; every plane rotation is a finite composition of these, including a right angle as two rotations through $\pi /4$. Lebesgue measure is invariant under every rotation.