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I was wondering what the definitions of one mapping dominating another in some general settings are?

A special case I inferred from Dominated Convergence Theorem is that: for mappings $f$ and $g$ from a set $X$ to $\mathbb{R}$, $f$ is called to dominate $g$, if for every $x \in X$, $f(x) \geq |g(x)|$.

Can we generalize their codomain from $\mathbb{R}$?

Generally, does it require the dominating function $f$ to be a nonnegative function?

Thanks and regards!

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Be aware of notion of majorating: the function $f$ is a majorant for $g$ (both real-valued) if $f(x)\geq g(x)$ for all values $x$ of the argument. Example: optimal stopping and optimal control theory. – Ilya Feb 22 '12 at 19:39
@Ilya: Thanks! Good to know. – Tim Feb 22 '12 at 19:42

A boring generalization: $\|f\|\ge \|g\|$ for maps into a normed space. This works for convergence of integrals, too.

More interesting example: subordination of mappings from the unit disk $D\subset \mathbb C$ into $\mathbb C$. Namely, $g$ is subordinate to $f$ if it factors as $g =f\circ\varphi $ where $\varphi :D\to D$ is injective and fixes $0$. (All maps are holomorphic). A classical early result on this subject is Littlewood's subordination theorem.

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