Is Young's inequality true without the continuity assumption?

Question

If $f:\mathbb{R} \to \mathbb{R}$ is a strictly increasing function, $f(0) =0$, $a>0$ and $b>0$, then $\int^a_0 f+ \int^b_0 f^{−1} ≥ ab$. Is this true without continuity assumed?

Answer 1

Here is a possible generalization. If $f$ is right-continuous on $[0,\infty)$ and $\lim_{x\to\infty}=\infty$, then $$ \bbox[cornsilk,5px] { \int_0^a f(x)\,dx + \int_0^b f^{-}(y)\,dy\ge ab } $$ for all $a,b\ge 0$, where $$ f^{-}(y):=\sup\{x\ge 0:f(x)\le y\} $$ is the generalized inverse of $f$.

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To see this, define \begin{align} A&:=\{(x,y):x\in[0,a],y\in[0,b]\},\\[5pt] B&:=\{(x,y):x\in[0,a],y\in[0,f(x)]\}, \quad\text{and}\\[5pt] C&:=\{(x,y):y\in[0,b],x\in[0,f^{-}(y)]\}. \end{align} Then, since $A\subseteq B\cup C$, $$ m(A)\le m(B)+m(C), $$ where $m$ is the Lebesgue measure on $\mathbb{R}^2$ (note that $A, B$, and $C$ are measurable sets).