Show that $V_G: L^2(G\times G, \mu \times \mu)\to L^2(G\times G, \mu \times \mu)$ defined by $V_G(f)(x,y) = f(xy,y)$ is well-defined Let $G$ be a locally compact Hausdorff group and let $\mu$ be a right Haar measure on $G$. Then $\mu\times \mu$ (the Radon product of measures) is a right Haar measure on $G \times G$ and we can define the operator
$$V_G: L^2(G\times G, \mu \times \mu)\to L^2(G\times G, \mu \times \mu)$$
by $$V_G(f)(x,y) = f(xy,y)$$
where $f$ is a measurable square integrable function on $G \times G$ and $x,y \in G$.
Is it true that $V_G$ is well-defined? I.e. if $f= g$ almost everywhere, then is it true that
$V_G(f)= V_G(g)$ almost everywhere?
Let $f \in C_c(G \times G)$. By proposition 7.22 of Folland's book "Real analysis", we can apply a version of Fubini's theorem without any assumptions on the $\sigma$-finiteness of the spaces. We obtain
\begin{align*}\int_{G\times G} |V_G(f)(x,y)|^2 (\mu \times \mu)(dx,dy) &= \int_{G \times G} |f(xy, y)|^2(\mu\times \mu)(dx,dy)\\
&= \int_G \left(\int_G |f(xy,y)|^2\mu(dx)\right) \mu(dy)\\
&= \int_G \left(\int_G |f(x,y)|^2 \mu(dx)\right)\mu(dy)\\
&= \int_{G \times G} |f(x,y)|^2 (\mu\times \mu)(dx,dy)\end{align*}
so it follows that $\|V_G(f)\|_2 = \|f\|_2$ for $f \in C_c(G \times G)$. We thus obtain an isometric mapping
$$V_G: C_c(G \times G)\to L^2(G \times G)$$
so at least the mapping is well-defined on $C_c(G \times G)$. Somehow I want to invoke density of $C_c(G \times G)$ in $L^2(G \times G)$ to conclude.

The above question is based on the following fragment in Timmerman's book "An invitation to quantum groups and duality":


 A: I would like to demur from the comments, and say that actually everything works fine, with appropriate (but standard) definitions.  The original question referenced Folland's Real Analysis book, so I'll make references to that to start with.
The first "problem" is that with $\mathcal B_G$ the $\sigma$-algebra of Borel sets on $G$, we don't necessarily have that $\mathcal B_G \times \mathcal B_G = \mathcal B_{G\times G}$, unless $G$ is second countable.  The standard way to "fix" this is to define a Radon measure on $G\times G$ to satisfy
$$ \int_{G\times G} f(x,y) \ d(\mu\hat\times\mu)(x,y) = \int_G \int_G f(x,y) \ d\mu(x) \ d\mu(y) $$
for all continuous compactly supported $f$.  (So $\mu\hat\times\mu$ is built from the Riesz representation theorem, and not directly as a product measure).  See Section 7.4 in Folland.
When $\mu$ is $\sigma$-finite, then essentially everything works as we might hope: Theorem 7.27 in Folland is the statement of the Fubini--Tonelli Theorem, showing for example that if $f\in L^1(\mu\hat\times\mu)$ then $x\mapsto f(x,y)$ is in $L^1(\mu)$ for almost all $y$, and
$$ \int f d(\mu\hat\times\mu) = \int\int f(x,y) \ d\mu(x) \ d\mu(y), $$
(and the other way around).  This formula also holds for any positive, Borel measurable function on $G\times G$ (and so $\mu\hat\times\mu$ extends the usual meaning of $\mu\times\mu$).
Thus, given $f\in L^2(\mu\hat\times\mu)$ (meaning that I take any function $f:G\times G\rightarrow\mathbb C$ with $\int |f|^2 < \infty$ and identify $f$ with its equivalence class) then as $\alpha:G\times G\ni (x,y) \mapsto (xy,y)\in G\times G$ is a continuous homeomorphism, $f\circ\alpha$ is Borel, and by Fubini--Tollelli applied to $|f\circ\alpha|^2$, we can invoke right-invariance:
\begin{align*}
\int |f\circ\alpha|^2 \ d(\mu\hat\times\mu)
= \int \int |f(xy,y)| \ d\mu(x) \ d\mu(y)
= \int \int |f(x,y)| \ d\mu(x) \ d\mu(y)
= \int |f|^2 \ d(\mu\hat\times\mu)
\end{align*}
So $V$ is an isometry (and then clearly a unitary, as $V^{-1}=V^*$ has a similar form).
(To concretely answer the original question, if $f=0$ almost everything for $\mu\hat\times\mu$ then the same is true of $f\circ\alpha$, and so $V$ is indeed well-defined.)
To show that $L^2(\mu\hat\times\mu) = L^2(\mu)\otimes L^2(\mu)$ one uses that $C_c(G\times G)$ is dense is both, and that the natural inner-products agree.  (See Proposition 7.9 in Folland).
Finally, what about the non $\sigma$-finite case?  This is dealt with by the "Interlude", section 2.3 in Folland's Harmonic Analysis book.  Fubini--Tonelli still holds if we can work in a $\sigma$-compact subset of $G\times G$.  We are only interested in, say, $L^p$ spaces, and then $f\in L^p$ will always vanish outside of a $\sigma$-compact subset.  Indeed, things are very nice for a locally compact group $G$ as then there is a $\sigma$-compact open and closed subgroup $G_0$, and we can write $G$ as the product of cosets of $G_0$.

A slightly different perspective if afforded by Cohn's book Measure Theory see Section 9.4 for convolution.  In particular, Lemma 9.4.3 exactly considers the analogue for the left Haar measure.  Cohn's presentation is slightly slicker as he proves Theorem 7.6.7 which is a version of Fubini for arbitrary Radon measures, and then Lemma 9.4.2 shows that when applied to a Haar measure, the hypotheses are always satisfied.
