Let $B_t$ be a $d$-dimensional Brownian motion. Define $P_t u(x):= Eu(x+B_t)$.
This is part of a solution to a problem from Schilling's Brownian Motion. Here, how is the integrand $|Eu(x+B_t)-u(x)|^p$ bounded by $2^p \Vert u\Vert_{L^p}^p$? Also, why is this continuous as a function of $t$? Finally, how do we know that $\Vert u(\cdot + B_t) - u\Vert_{L^p}^p$ is finite?
At the very beginning of the solution it is stated that the mapping
$$y \mapsto \|u(\cdot+y)-u\|_{L^p}$$
is a continuous function. Since the Brownian motion has (almost surely) continuous sample paths, this clearly implies that
$$t \mapsto \|u(\cdot+B_t)-u\|_{L^p}$$
is (almost surely) continuous as a composition of continuous mappings.
Regarding the upper bound: You are confusing things. The author wants to apply domainted convergence to
$$\mathbb{E}(\|u(\cdot+B_t)-u\|_{L^p}^p)$$
which means that we have to find a nice upper bound for the integrand
$$\|u(\cdot+B_t)-u\|_{L^p}^p.$$
Using that
$$|x+y|^p \leq (2\max\{|x|,|y|\})^p \leq 2^p (|x|^p+|y|^p)$$
we find
$$\begin{align*} \|u(\cdot+B_t)-u\|_{L^p}^p &= \int |u(x+B_t)-u(x)|^p \, dx \\&\leq 2^p \int |u(x+B_t)|^p \, dx + 2^p \int |u(x)|^p \, dx \\ &= 2^{p+1} \int |u(y)|^p \, dy =: c. \end{align*}$$
The right-hand side is nicely integrable upper bound for the integrand, and so we may apply the dominated convergence theorem.