The resource bounded measure theory developed by Jack Lutz is a great area for people who have background in analysis to work on. The original paper
Almost everywhere high nonuniform complexity,
Jack H. Lutz, Journal of Computer and System Sciences, 1992.
generalize the notion of Lebesgue measure into complexity classes, and many following works can be found on the internet.
Intuitively, consider the $\mathsf{P}$ vs $\mathsf{NP}$ problem. If we can define (yes we can) a measure on complexity classes with respect to a large class, say $\mathsf{ESPACE} = \mathsf{DSPACE}[2^{O(n)}]$, and prove that the measure of $\mathsf{P}$ is smaller than the measure of $\mathsf{NP}$, then $\mathsf{P} \neq \mathsf{NP}$. Moreover, we can prove statement like "almost all functions in $\mathsf{ESPACE}$ need $\Omega(2^n/n)$ gates", which extends Shannon bound to a restricted class $\mathsf{ESPACE}$.