Concentration without first moment The weakest concentration inequality I know of is markov's inequality:
$$\mathbb P (X \geq t) \leq \frac{\mathbb E X}{t}$$
where $X$ is a nonnegative random variable with first moment $\mathbb E X$ and $t > 0$. Of course, this bound assumes that the first moment exists.
Is there a weaker and still useful concentration inequality than the markov inequality? Does it even make sense to talk about concentration when the first moment does not exist?
For example in the specific case of a cauchy, I know that the measure concentrates around $0$, but I don't know of a useful bound on the measure far away from this concentration point that applies to general random variables without finite first moments.
EDIT: naively, I could imagine the existence of finite "partial moments" implying weaker inequalities. By partial moments I mean: $\mathbb E X^p$ for $p < 1$ (although I dont think negative $p$ could give us concentration type properties).

On $\mathbb{R}$ (and Polish spaces in general) all probability measures are tight, so some kind concentration inequality always exists and can be derived directly by the tightness property. (See https://en.wikipedia.org/wiki/Tightness_of_measures)
I would also add that for any bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$ and any ranom variable $X$ on $\mathbb{R}$ it always holds that $\mathbb{E}f(X) < \infty $.
If $f$ is also strictly increasing then we can always use Markov's inequality to derive the concentration bound
$$ P(X>f^{-1}(t))\leq \frac{\mathbb{E} f(X)}{t}.$$
