I want to show that
$$f(x) \in \mathcal{M} (\Bbb{R}, \mathfrak{B}) \Leftrightarrow g_y(x):=f(x+y) \in \mathcal{M} (\Bbb{R}, \mathfrak{B}), \ \ \forall y \in \Bbb{R}$$
'$\in \mathcal{M}$' means measurable
and that then
$$\int f \mathrm{d}\lambda = \int g_y \mathrm{d} \lambda$$ if both sides exist.
For the first part: Let $[f<c] := \{x|f(x)<c\}$
$$x \in [g_y<c] \Leftrightarrow = g_y(x) <c \Leftrightarrow f(x+y)<c \Leftrightarrow (x+y) \in [f<c] \Leftrightarrow x \in [f<c]-y$$
Therefore $$[g_y<c]=[f<c]-y$$
Now, because the borel-sigma-algebra is translation-invariant
$$[f<c] \Rightarrow [f<c]-y \Rightarrow [g_y<c]$$ is measurable.
Is that correct? And how do I show $\int f \mathrm{d}\lambda = \int g_y \mathrm{d} \lambda \ $ if both sides exist?