Translation Invariance of Lebesgue Measure and the Lebesgue Integral

For a subset $A$ of the real line $\mathbb{R}$ and a real number $t$, write $A+t=\{x+t:x\in A\}$. Let $\lambda^{*}$ be the Lebesgue outer measure, $\lambda$ Lebesgue measure, and $\mathcal{B}(\mathbb{R})$ the Borel $\sigma$-algebra. Then for every real $t$:

1. (Sets) For every $A\subseteq\mathbb{R}$, $\lambda^{*}(A+t)=\lambda^{*}(A)$. For every Borel set $B$, the translate $B+t$ is Borel and $\lambda(B+t)=\lambda(B)$.

2. (Integrals) For every measurable function $f:\mathbb{R}\to\mathbb{R}$ (with respect to $\mathcal{B}(\mathbb{R})$) with $f\ge0$, the function $x\mapsto f(x+t)$ is measurable and, with the integral of nonnegative measurable functions,