The following is used on page 118 of Karl Petersen's Ergodic theory book.
Let $T_t$ be a measure preserving flow and let $f(x)=\int_{-\infty}^{\infty}f_0(T_s(x))\phi(x)ds$, where $f_0$ is a bounded function and $\phi$ is differentiable with compact support. We want to show that the set generated by the functions that look like $f(x)$ along with the invariant functions, call this set $D$, is dense in $L_1$.
The author claims that it is enough to show that for every bounded $h$ we have $\int hf d \mu =0$ for all such $f$ mentioned above. The justification is that by the Hahn-Banach theorem, any function in $D$ can be separated by a linear functional from any function not in it.
I do not understand the claim $D$ is dense if $\int hf = 0$ for $f \in D$ implies h is invariant.
Not sure why there's a downvote. I figured out a solution but I had to use the big rigs to solve it.
Suppose not. Then there is a bounded $h$ such that $\int hf=0$ and a $g$ not in our subspace such that $ \int gh \neq 0$. We will show first that $h$ is invariant. Notice that after changing coordinates and setting $f_0=1$ our condition implies that $0=\int_{- \infty}^{\infty} h(T_{-s}x) \phi(s) ds$ for all $\phi$ that are differentiable with compact support. If we let $\phi$ be a smooth version of the function $\frac{1}{m(I)}\chi_I - \frac{1}{m(J)} \chi_{J}$ and shrink the intervals, the local ergodic theorem will give us that $h$ is constant on almost every orbit. Hence $h$ is invariant. But the definition of $h$ implies that $\int h^2 = 0$. This implies that $h=0$ a.e. and that we cannot have $\int hg \neq 0$.