Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|\_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of Lebesgue measurable functions $f:\mathbb R^d \to \mathbb R$ such that
$$
\|f\|\_{\tilde L^p} := \sup\_{x \in \mathbb R^d} \|1\_{B(x, 1)} f\|\_{L^p} < \infty,
$$
where $B(x, 1)$ is the open unit ball centered at $x$. I have [verified](https://math.stackexchange.com/questions/4750669/the-space-lp-textloc-mathbb-rd-is-complete-w-r-t-the-norm-f) that $(\tilde L^p (\mathbb R^d), \|\cdot\|\_{\tilde L^p})$ is complete. Let $(x\_m)$ be a countable dense subset of $\mathbb R^d$. The sphere has Lebesgue measure $0$. Then by dominated convergence theorem (DCT),
$$
\|f\|\_{\tilde L^p} = \sup\_{m \in \mathbb N} \|1\_{B(x\_m, 1)} f\|\_{L^p}
\quad \forall f \in \tilde L^p (\mathbb R^d).
$$


I would like to ask if a version of DCT holds for $L^p (\mathbb R^d)$, i.e.,



> 
> Let $f\_n, g \in \tilde L^p (\mathbb R^d)$ such that $|f\_n| \le g$ a.e. for all $n$. If there is $f:\mathbb R^d \to \mathbb R$ such that $f\_n \to f$ a.e., then $f \in \tilde L^p (\mathbb R^d)$ and $\|f\_n-f\|\_{\tilde L^p} \to 0$.
> 
> 
> 


Thank you so much for your elaboration!



Is there a version of dominated convergence theorem for local $L^p$ spaces?
A counterexample is given by $f\_n=1\_{(n,\infty)}$, $g=1$, and $f=0$.


Then all the conditions on $f\_n,g,f$ hold, but $\|f\_n-f\|\_{\tilde L^p} \not\to0$.


