dominated convergence for functions $\mathbb R^n\to\mathbb R^m$?

Question

I do know the dominated convergence theorem for functions $f:\mathbb R^n\to\mathbb R$.

Now let $U\subset\mathbb R^n$ and $f: U\to\mathbb R^m$. Is there any dominated convergence theorem for 'vectorial' functions?

Clearly one could integrate each component and apply the dominated convergence theorem for each component but can you apply it too without using this fact?

Especially what a about the dominated function, can you use a norm $|\cdot|$ and somenthing like $|f|\leq |g|$?

Answer 1

One approach is to note that if $x_k \in \mathbb{R}^n$, then $x_n \to x$ iff $\phi(x_k) \to \phi(x)$ for any linear functional $\phi$ (this works in $\mathbb{R}^n$ since strong and weak convergence coincide).

In particular, given some $f_n, f, g$ with $f_n(x) \to f(x)$ and $\|f_n\| \le g$, then for a linear functional $\phi$, we have $|\phi(f_n)| \le \|\phi\| \|f_n\| \le \|\phi\|g$, and so $\int \phi(f_n) \to \int \phi(f)$.

Since $\phi(\int f_n) = \int \phi(f_n), \phi(\int f) = \int \phi(f)$, we see that $\phi(\int f_n) \to \phi(\int f)$ for all $\phi$, and hence $\int f_n \to \int f$.

Answer 2

The dominated convergence theorem applies to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable with $\|f_n(x)\|\leq g(x)$

The assumption of convergence almost everywhere can be weakened to require only convergence in measure.