# $f,g$ continuous maps. $f = g$ a.e. equal, is $f = g$ on a locally compact group?

In this question (functions $f=g$ $\lambda$-a.e. for continuous real-valued functions are then $f=g$ everywhere) it is stated that if $f,g : \mathbb{R} \to \mathbb{R}$ are continuous and $f = g$ a.e. equal then in fact $f(x) = g(x)$ on all of $\mathbb{R}$.

What if $f,g : G \to \mathbb{C}$ (continuous maps) where $G$ is a locally compact group? Does it still hold that $f = g$ a.e. equal implies $f(x) = g(x)$ for all $x\in G$?

If you are talking about the Haar-measure on $G$, this is true.

For if there was a nonempty open set $\emptyset \neq U \subset G$ with $\mu(U) = 0$, we can assume (by translation) w.l.o.g. that $e \in U$ (the identity of $G$).

Then for every compact $K \subset G$, we could cover $K$ by finitely(!) many of the translates $xU$ for $x \in K$ (here, we use that $K$ is compact).

This implies $\mu(K) = 0$ for every(!) compact $K \subset G$.

By inner regularity of the Haar measure, i.e. by

$$\mu(V) = \sup\{\mu(K) \mid K \subset V \text{ compact} \},$$

we conclude $\mu(V) = 0$ for all open subsets $V \subset G$. By outer regularity, we get $\mu \equiv 0$, a contradiction.

This shows that every nonempty open set has positive measure.

By continuity of $f,g$ we know that

$$U := \{ x \in G \mid f(x) \neq g(x) \}$$

is open. Thus it is either empty (i.e. $f \equiv g$), or has positive measure (i.e. NOT $f=g$ a.e.).

• Thank you, I didn't expect a proof to take this route at all, but it makes sense when you explain it. Was wondering for a moment how you could know that $U$ must be open, but this is of course due to the uniqueness of limits. :)
– zo0x
Commented Jun 11, 2014 at 10:56
• You could also use that $U = (f-g)^{-1}(\Bbb{C}^\ast)$ is open as the inverse image of an open set under the continuous function $f-g$. Commented Jun 11, 2014 at 11:44
• That makes sense. I assume $\mathbb{C}^\ast = \mathbb{C}\backslash\{0\}$? Is this a standard notation?
– zo0x
Commented Jun 11, 2014 at 12:37
• Yes, it is standard notation for the group of units of a ring (or a field in this case). See en.wikipedia.org/wiki/Unit_%28ring_theory%29 Commented Jun 11, 2014 at 12:56