Can an improper prior distribution be informative? I have just worked through an example where, with an improper prior, the bayesian estimator equals the maximum likelihood estimator, leading me to believe that improper priors are uninformative. But must this be true?
 A: The prior which is uniform over $[m,\infty)$ is improper, but informative: it contains the information that the value is at least $m$.
A: 
...example where, with an improper prior, the Bayesian estimator
  equals the maximum likelihood estimator...

There is a fundamental issue with this property, namely that it depends on the parameterisation of the sampling model. Indeed, if
$$\hat\theta^\text{MAP}=\arg\max_\theta L(\theta|x)\pi(\theta)=\arg\max_\theta L(\theta|x)=\hat\theta^\text{MLE}\tag{1}$$
consider the change of parameterisation $\eta=h(\theta)$, where $h$ is a bijection. Then the prior on $\eta$ is
$$\tilde\pi(\eta)=\pi(h^{-1}(\eta))\ \underbrace{\left|\dfrac{\text{d}h^{-1}(\eta)}{\text{d}\eta}\right|}_\text{Jacobian}$$
and there is no reason for
$$\hat\eta^\text{MAP}=\arg\max_\eta L(\eta|x)\tilde\pi(\eta)=\arg\max_\eta L(\eta|x)=\hat\eta^\text{MLE}$$
to hold, even though $$\hat\eta^\text{MLE}=h(\hat\theta^\text{MLE})$$
because of the Jacobian. MAP estimators are not invariant by reparameterisation.
As an additional remark, that a particular prior leads to this (dubious) identity is not one of the definitions (here, there, or there) of non-informative priors, since, for one thing, being non-informative is a property that depends on both the data (I assume (1) holds for a particular $x$) and the parameterisation. Furthermore, improper priors should not be assimilated to non-informative priors.
