When is $\int f(x+h) \, dx \neq \int f(x) \, dx$? In Stein we have that if $f$ is a function of "moderate decrease" then the integral is translation invariant. For all $h \in \mathbb{R}^2$:
$$ \int_{\mathbb{R}^2} f(x+h) \, dx = \int_{\mathbb{R}^2} f(x) \, dx $$
Stein's textbook on Fourier analysis suggests two things:

*

*$f$ is continuous on $\mathbb{R}^2$


*$\displaystyle \sup_{x \in \mathbb{R}^2} |x|^2 |f(x)| < \infty$ ("moderate decrease", more casually $f(x) \propto \frac{1}{|x|^2}$.
The integral over $\mathbb{R}^2$ is defined as the limit of the Riemann integrals over larger and larger areas:
$$ \int_{\mathbb{R}^2} f(x) \, dx = \lim_{N \to \infty} \int_{Q_N} f(x) \, dx $$
The large area here is just a square oriented with sides parallel to the coordinate axis:

*

*$Q_N = \big\{ x \in \mathbb{R}^2 : |x_1| \leq \frac{1}{2}N \text{ and } |x_2| \leq \frac{1}{2}N \big\}$
This is the Analysis textbook so there is a similar objection for the rescaling by a constant and for rotations.

*

*When is $\displaystyle \delta^2 \int_{\mathbb{R}^2} f(\delta x) \, dx \neq \int_{\mathbb{R}^2} f(x) \, dx$ ?


*When is $\;\;\;\;\displaystyle \int_{\mathbb{R}^2} (f \circ R) (x) \neq \int_{\mathbb{R}^2} f(x) \, dx $ for $R \in \text{SO}_2(\mathbb{R})$ the rotation group;  Euclidean plane is invariant under rotations.
 A: well if we look at:
$$\int\limits_{\mathbb R^2}f(x+h)\,dx$$
making the linear transformation $u=x+h$ we get $x\in(-\infty,\infty)\mapsto u\in(-\infty,\infty)$ and so we get:
$$\int\limits_{\mathbb R^2}f(x+h)\,dx=\int\limits_{\mathbb R^2}f(u)\,du\equiv \int\limits_{\mathbb R^2}f(x)\,dx$$
and so the two are equal, or the integral is independent of $h$.

As for a function of "moderate decrease", I would interpret that as:
$$O\left(\frac1{f(x)}\right)>kx^2\qquad k>0$$

The conditions stated are not to do with the integral being independent of $h$, but rather that it exists so that when it undergoes a linear transformation it is invariant.
A: The conditions do have to do with the integral being independent of $h$ and in fact are not sufficient to show existence of the integral.
In particular, consider shifting $Q_N$ be $h$. This gives you two squares which overlap in a square of side length at least $N-|h|$. The size of the non-overlap region is at most $N^2-(N-|h|)^2=2N|h| + |h|^2 = O(N)$ and the non-overlap region is at least distance $N-|h|=O(N)$ from the origin. Let's call this region $R_{N,h}$. Then, for any $N$:
$$|\int_{Q_N} f(x+h) dx - \int_{Q_N} f(x) dx|=|\int_{Q_N} f(x+h) dx - \int_{Q_N} f(x) dx|$$
$$\leq |\int_{R_{N,h}} |f(x)| dx$$
$$\leq (2N|h| + |h|^2) \cdot \frac{c}{(N-|h|)^2} \sim N \cdot \frac{1}{N^2}=\frac{1}{N}$$
Thus, if the integral exist, then it's invariant to shifting.
You can similar show it's invariant to scaling (since that's just changing $N$). It is not invariant to rotations, since the overlap between squares is $N^2$, so you'd need either abosolute convergence or a stronger decrease condition like $1/x^{2+\epsilon}$ convergence.
His moderate decrease condition is not sufficient for convergence. For example, $1/(1+|x|^2)$ satisfies his condition, but diverges over the plane. You could also construct functions that don't absolutely converge but satisfy his convergence. I think $\sin(\max(|x|,|y|))/(1+\max(|x|,|y|)^2)$ is one such function.
