  Let $X \subset \mathbb{R}^N$ be a Borel set, $\mu$ a Borel probability
measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in
\mathbb{N}$ greater than the (Hausdorff) dimension of $X$ and assume that the
set of $p$-periodic points has dimension smaller than $p$ for $p=1, \ldots,
k-1$. We prove that for a typical polynomial perturbation $\tilde{h}$ of a
given Lipschitz map $h : X \to \mathbb{R}$, the $k$-delay coordinate map $x
\mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ is
injective on a set of full measure $\mu$. This is a probabilistic version of
the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We
also provide a non-dynamical probabilistic embedding theorem of similar type,
which strengthens a previous result by Alberti, B\"{o}lcskei, De Lellis,
Koliander and Riegler. In both cases, the key improvements compared to the
non-probabilistic counterparts are the reduction of the number of required
measurements from $2\dim X$ to $\dim X$ and using Hausdorff dimension instead
of the box-counting one. We present examples showing how the use of the
Hausdorff dimension improves the previously obtained results.
