I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem:
For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time point $t$. Let $p \geq 1$ and $q$ be its Holder's conjugate. Suppose that there exists a nonnegative function $K(t)$ such that $$ \big| \mathbb{E} [ f( X_t) ] \big| \leq K(t) \| f \|_p .$$ Then $$ \| u_t \| _q \leq K(t).$$
As far as what I know, the Riesz's theorem merely tells us that for a compact Hausdorff space $X$, for any bounded linear functional $\Lambda \in C(X)$, we can correspond it to a Borel measure $\mu$ through the Lebesgue integral. How does this conclude this result? Or is there an alternative version of this theorem which might be helpful in showing this?
As Wikipedia points out, there are several results called Riesz (representation) theorem. A common feature of some of them is that on some spaces, every linear functional is integration against something. The result that the author uses here is the representation of linear functionals on $L^q$: they are integrals against $L^p$ functions. More precisely, all that's needed here is the equality case of Hölder's inequality: $$ \|u_t\|_q = \sup\left\{\int f u_t : \|f\|_p\le 1\right\}\tag{1} $$
Since $E[f(X_t)] = \int f(\omega)u_t(\omega)$, the result follows.