Translation property in $L^1(\mathbb{R})$ space

Question

Let $g(x)$ be a bounded measurable functions on $\mathbb{R}$, and $f(x)$ be in $L^1(\mathbb{R})$.

Notation: $\int_\mathbb{R} h(x)dx=\ $the integration of measurable function $h$ over $\mathbb{R}$

I would like to ask if the following statements holds.

1. $\forall c\in\mathbb{R}\ \int_\mathbb{R}f(x+c)dx=\int_\mathbb{R}f(x)dx$
2. $\lim_{t\rightarrow0} \int_\mathbb{R} g(x)|f(x+t)-f(x)|dx=0$

Thank you very much!

Answer 1

Sketch: Certainly $1.$ holds, because Lebesgue measure is translation invariant. For 2., try to prove it first for compactly supported continuous functions. This follows nicely from uniform continuity. To finish, use the fact that such functions are dense in $L^1.$