We know that $l_i=\log \frac{1}{p_i}$ is the solution to the Shannon's source compression problem: $\arg \min_{\{l_i\}} \sum p_i l_i$ where the minimization is over all possible code length assignments $\{l_i\}$ satisfying the Kraft inequality $\sum 2^{-l_i}\le 1$.
Also $H(p)=\log \frac{1}{p}$ is additive in the following sense. If $E$ and $F$ are two independent events with probabilities $p$ and $q$ respectively, then $H(pq)=H(p)+H(q)$.
As far as I know, mainly for these two reasons $H(p)=\log \frac{1}{p}$ is considered as a measure of information contained in a random event $E$ with probability $p>0$.
On the other hand, if we average the exponentiated lengths, $\sum p_i2^{tl_i}, t>0$, subject to the same Kraft inequality constraints, the optimal solution is l_i=\log \frac{1}{p_i'} where p_i'=\frac{p_i^{\alpha}}{\sum_k p_k^{\alpha}}, \alpha=\frac{1}{1+t}, called Campbell's problem.
Now H_{\alpha}(p_i)=\log \frac{1}{p_i'} is also additive in the sense that $H_{\alpha}(p_i p_j)=H_{\alpha}(p_i)+H_{\alpha}(p_j)$. Moreover $H_{\alpha}(1)=0$ as in the case of Shannon's measure.
Also note that, when $\alpha=1$, $H_1(p_i)=\log \frac{1}{p_i}$ we get back Shannon's measure.
My question is, are these reasons suffice to call H_{\alpha}(p_i)=\log \frac{1}{p_i'} a (generalized) measure of information?
I don't know whether the dependence of measure of information of an event also on the probabilities of the other events make sense.