Supremum outside or inside an integral

Question

Let $X$ be a Banach space and let $\alpha:\mathbb{R}\to X$ be a bounded function: for all $x\in\mathbb{R}$ $$|\alpha(x)|\le M$$ and $f:\mathbb{R}\times X\to X$ is two variables function. We know that $$\int_a^b|f(x,\alpha(x))|dx\leq \int_a^b\sup_{|y| \leq M} |f(x,y)|dx.$$ Does the following inequality holds $$\int_a^b|f(x,\alpha(x))|dx\leq \sup_{|y| \leq M} \int_a^b |f(x,y)|dx \ \ ?$$ I think it holds if the two variable function $f$ has a separated form, i.e $f(x,y)=f_1(x)f_2(y)$. Does the inequality hold in general ?

Answer 1

This inequality is not true in general. Consider $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by $$f(x,y)=1_{\{y\}}(x),$$ Then, for any $a<b$, $$\int_a^b |f(x,y)|dx=0\:\:\Rightarrow\:\: \sup_{|y|\leq M}\int_a^b |f(x,y)|dx=0,$$ but by defining $\alpha(x)=x1_{[0,1]}(x)$ we get for $a\leq 0$ and $b\geq 1$, $$\int_a^b|f(x,\alpha(x))|dx=1.$$