# If $f$ measurable positive, is there a sequence of simple function that converge decreasly to $f$?

A famous theorem says that if $$f\geq 0$$ is measurable, then there is an increasing sequence of simple function $$(\varphi _n)$$ s.t. $$\varphi _n\nearrow f$$. Then we can define $$\int_{\mathbb R}f:=\lim_{n\to \infty }\int_{\mathbb R}\varphi _n.$$

Now my question is : in these conditions, (i.e. $$\geq 0$$ and measurable), is there a decreasing sequence of simple function $$(\psi_n)$$ s.t. $$\psi_n\searrow f$$ ?

Attempts

I would say no because otherwise we could define $$\int f=\lim_{n\to \infty }\int_{\mathbb R}\psi_n,$$ and monotone convergence theorem (MCT) would work as well. But I know that MCT doesn't hold, so I guess this result is not correct. Does someone has a counter example ? And if not, does this result hold ?

The answer is no. But your explanation why makes no sense at all: The fact that there would be problems using $$(\psi_n)$$ to define $$\int f$$ does not show that $$(\psi_n)$$ does not exist. You say "no because otherwise[...], and monotone convergence theorem (MCT) would work as well. But I know that MCT doesn't hold," This is unclear. MCT does hold - I can prove it.
• What do you mean by MCT does hold ? That if $\psi_n\searrow f$ then $\lim_{n\to \infty }\int \psi_n=\int f$ ? Mar 3 '19 at 9:28