Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 4 / 2

The indicator functions satisfyTherefore ${\phi }_1,{\phi }_2\in L^2[0,1]$.

For every $h\in L^2[0,1]$, the continuous linear functional induced by the inner product givesThis is a normal random variable because it is a linear combination of independent normal random variables. Hence the law of $U$ is a Gaussian measure. Its mean is zero, and independence together with $\mathrm{Var}({\xi }_i)=1/2$ givesThus its covariance operator of a Gaussian measure is

The supports of ${\phi }_1$ and ${\phi }_2$ are disjoint, so they are orthogonal vectors. Substitution in the covariance formula givesandTherefore the associated eigenvalues are