Problem 1: Let $f\in L^1(\mathbb{R})$, show $\int_\mathbb{R}f(t)dt=\int_{\mathbb{R}}f(x+t)dt,\forall x\in (-\infty, \infty)$.
Problem 2: Let $f\in L^1(\mathbb{R})$,show $\displaystyle \lim_{t\rightarrow 0}\int_{\mathbb{R}}|f(x+t)-f(x)|dx=0$.
My try for Problem 1:
Consider $\displaystyle \int_{\mathbb{R}}f(t)dt=\lim_{n\rightarrow \infty}\int_{-n}^nf(t)dt=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f_n(t)dt$, where $f_n(t)=f(t)\chi_{[-n,n]}$.
And since $dt=d(x+t)$, we have $\displaystyle \int_{-n}^nf(x+t)dt=\int_{-n+x}^{n+x}f(t)dt$.
So for each $x\in (-\infty,\infty)$, $$ \begin{split} \int_{\mathbb{R}}f(t)-f(t+x)dt &=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f(t)(\chi_{[-n,n]}-\chi_{[-n+x,n+x]})dt\\ &=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt\\ &=\lim_{n\rightarrow \infty}\lim_{k\rightarrow \infty}\int_\mathbb{R}\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt\\ &=\lim_{k\rightarrow \infty}\lim_{n\rightarrow \infty}\int_\mathbb{R}\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt \end{split} $$ where $\varphi_k(t)$ is simple function pointwise converges to $f(t)$, simple function vanish out of set of finite measure, so $\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})$ vanishes when $n$ big enough(is this right?)
Another try:
Since $f\in L^1$, $f$ is bounder a.e. so can be approximated by simple functions, and $\chi_{A+t}(x)=\chi_A(x-t)$,thus $$\int f(t)=\lim_n\int\varphi_n=\lim_n \sum_{i=1}^nc_i\mu(E_i)=\lim_n \sum_{i=1}^nc_i\mu(E_i-x)=\int f(t+x)$$
For problem 2, I guess they share the same method.
Any help or correction about these two problems? Thanks.
In your solution, it's not clear to me why you can just assert that $\int_{-n}^nf(x+t)\;dt=\int_{-n+x}^{n+x}f(t)\;dt$. This seems to be essentially equivalent to what you're trying to prove.
Instead, I'd proceed in the following way: first prove that $\int_{\mathbb{R}}f(x+t)\;dt=\int_{\mathbb{R}}f(t)\;dt$ when $f=\chi_E$ is a characteristic function (this should be immediate from the translation invariance of Lebesgue measure). Then prove it for simple functions by the linearity of the integral, then for non-negative functions by approximating with simple functions and using the monotone convergence theorem, and finally for arbitrary $L^1$ functions by linearity again.
For part 2, I suggest using the fact that the set of continuous functions with compact support is dense in $L^1(\mathbb{R})$, together with the fact that a continuous compactly supported function is uniformly continuous.