Measures $\mu$ such that $\mu(a+A)\leq c\ \mu(A)$ Let $\mu$ be a positive measure on $\mathbb{R}$ such that $\mu[a,b]<+\infty$, for all $a,b\in\mathbb{R}$ and $\mu(\mathbb{R})=+\infty$. The set $a+A$ denotes the translation set of $A$ by a, i.e. $a+A=\{a+x, \text{with }x\in A \}$.
Consider the following hypotheses on the measure $\mu$:
(H0) For all $a\in \mathbb{R}$ and $A$ measurable set
$$\mu(a+A)=\mu(A).$$
(H1) $\exists c>0$ such that for all $a\in \mathbb{R}$ and $A$ measurable set
$$\mu(a+A)\leq c\ \mu(A).$$
Consider also the weaker hypothesis:
(H2) For all $a\in \mathbb{R}$, $\exists c_a>0$ such that for all $A$ measurable set
$$\mu(a+A)\leq c_a\ \mu(A).$$
We know that if $\mu$ satisfies (H0), then $\mu$ is nothing but the Lebesgue measure multiplied by a constant. What can we say about (H1) and (H2) ?
 A: Fix $\beta>0$ and $c \geq 1$. For any measurable function $f(x)$ that satisfies $1 \leq f(x) \leq c$ for all $x \in \mathbb{R}$, the measure $\mu(A) = \int_{x\in A} \beta f(x)dx$ satisfies H1. 
Below it is shown that this example structure captures every measure $\mu(A)$ that satisfies H1 together with the properties $\mu([0,1])<\infty$ and $\mu(\mathbb{R})>0$. 

Let $L(A)$ be the usual Lebesgue measure.  Let $\mu(A)$ be any measure on the same $\sigma$-algebra that satisfies $\mu([0,1])<\infty$ and $\mu(\mathbb{R})>0$. Suppose $\mu(A)$ satisfies H1 with respect to a particular number $c\geq 1$.  Define
$\beta = \inf_{a<b} \left[\frac{\mu([a,b])}{L([a,b])}\right]$, where the infimum is taken over all closed finite intervals $[a,b]$. 
You can prove that: 
(1) $0 < \beta < \infty$.  
(2) For any interval $[a,b]$ one has $\beta L([a,b])  \leq \mu([a,b]) \leq c\beta L([a,b])$. 
(3) By properties of inner and outer Lebesgue measure, property (2) implies $\beta L(A) \leq \mu(A) \leq c\beta L(A)$ for all Lebesgue measurable sets $A$. 
(4) Property (3) implies $\mu(A)$ is absolutely continuous with respect to the Lebesgue measure, and so (by Radon-Nikodym) there is a measurable function $g(x)$ such that $\mu(A) = \int_{x\in A} g(x)dx$ for all measurable sets $A$.  Define $f(x) = g(x)/\beta$.  
(5) You can show that $1 \leq f(x) \leq c$ almost everywhere (so just define it to be 1 at points where this is violated).  Then $\mu(A)$ has the form $\mu(A) = \int_{x\in A} \beta f(x)dx$ for a measurable function $f(x)$ that satisfies $1 \leq f(x)\leq c$ for all $x \in \mathbb{R}$.
