  Given strong uniqueness for an It\^o's stochastic equation, we prove that its
solution can beconstructed on "any" probability space by using, for example,
Euler's polygonal approximations. Stochastic equations in $\mathbb{R}^{d}$ and
in domains in $\mathbb{R}^{d}$ are considered. This is almost a copy of an old
article in which we correct errors in the original proof of Lemma 4.1 found by
Martin Dieckmann in 2013. We present also a new result on the convergence of
"tamed Euler approximations" for SDEs with locally unbounded drifts, which we
achieve by proving an estimate for appropriate exponential moments.
