  We extend Ghys' theory about semiconjugacy to the world of measurable
cocycles. More precisely, given a measurable cocycle with values into
$\text{Homeo}^+(\mathbb{S}^1)$, we can construct a
$\text{L}^\infty$-parametrized Euler class in bounded cohomology. We show that
such a class vanishes if and only if the cocycle can be lifted to
$\text{Homeo}^+_{\mathbb{Z}}(\mathbb{R})$ and it admits an equivariant family
of points.
  We define the notion of semicohomologous cocycles and we show that two
measurable cocycles are semicohomologous if and only if they induce the same
parametrized Euler class. Since for minimal cocycles, semicohomology boils down
to cohomology, the parametrized Euler class is constant for minimal
cohomologous cocycles.
  We conclude by studying the vanishing of the real parametrized Euler class
and we obtain some results of elementarity.
