Prove For $\epsilon >0$ exists a $g$ function bounded such that $g$ is integrable and $$\left|\int{f(x)}dx-\int{g(x)}dx\right|<\epsilon$$ where $f:\mathbb{R}\rightarrow\bar{\mathbb{R}}$ is a lebesgue integrable.
I think this problem can be solved using a Dominate convergence theorem but i don't have a clear idea of how solve this. can someone help me?
We know that $f<\infty$ a.e. Let $$g_n(x)=\begin{cases}f(x)&\text{if}\ |f(x)|\leq n,\\0&\text{otherwise},\end{cases}$$ then $g_n\to f$ a.e. Now use DCT to obtain $$\int g_n\,dx \to \int f\,dx.$$By the definition of limit, there exists $N$ such that the inequality in the post holds for all $n>N$. Let $g=g_{N+1}$ so $g$ is bounded by $N+1$.
Remark. In deed, we can prove that for any $\epsilon>0$ there exists a smooth function $g$ with compact support such that $$\left|\int f(x)\,dx-\int g(x)\,dx\right|<\epsilon,$$ where $f:\mathbb{R}\rightarrow\bar{\mathbb{R}}$ is a lebesgue integrable function.
See here: Smooth functions with compact support are dense in $L^1$
