Let
- $(E,\mathcal E)$ be a measurable space;
- $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
- $\mu$ be a finite measure on $(E,\mathcal E)$.
Assume $\mu$ is subinvariant with respect to $(\kappa_t)_{t\ge0}$. Let $f\in\mathcal L^p(\mu)$. Then, $$\mu(\kappa_t|f|^p)\le\mu|f|^p<\infty\tag1$$ and hence $$\kappa_t|f|^p<\infty\;\;\;\mu\text{-almost surely}\tag2$$ for all $t\ge0$.
Are we even able to show that $$\kappa_t|f|^p<\infty\;\;\;\text{for all }t\ge0\;\mu\text{-almost surely}\tag3?$$
Maybe we can use the contractivity of $(\kappa_t)_{t\ge0}$ to show some kind of monotonicity from which in turn we can derive $(3)$ ...