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We study the invariant subspaces generated by inner functions for a class of$\mathcal{P}^t(\mu)$-spaces which can be identified as spaces of analyticfunctions in the unit disk $\mathbb{D}$, where $\mu$ is a measure supported inthe closed unit disk and $\mathcal{P}^t(\mu)$ is the span of analyticpolynomials in the usual Lebesgue space $L^t(\mu)$. Our measures define a rangeof spaces somewhere in between the Hardy and the Bergman spaces, and ourresults are thus a mixture of results from these two theories. For a largeclass of measures $\mu$ we characterize the cyclic inner functions, and exhibitsome interesting properties of invariant subspaces generated by non-cyclicinner functions. Our study is motivated by a connection with the problem ofsmooth approximations in de Branges-Rovnyak spaces.

Inner functions, invariant subspaces and cyclicity in $\mathcal{P}^t(\mu)$-spaces