This question is on a point in D.R. Adams paper "A Sharp Inequality of J. Moser for Higher Order Derivatives". Precisely the lemma says:

Given $a(s,t)$ be a non negative measureable function on $(-\infty,\infty)\times [0,\infty)$ such that
$$
a(s,t)\leq 1\;\text{ when }\;0<s<t\label{1}\tag{1}
$$ and
$$
\sup_{t>0}\left(\int_{-\infty}^0+\int_t^{\infty} a(s,t)^q ds\right)^{\frac{1}{q}}=b<\infty\label{2}\tag{2}
$$
Then there is a constant $c_0=c_0(p,b)$ such that if $\phi\geq 0$ and
$$
\int_{\mathbb{R}}\phi(s)^p ds\leq 1\label{3}\tag{3}
$$
then
$$
\int_{0}^{\infty} e^{-F(t)} dt\leq c_0
$$
where
$F(t)=t-\left(\int_{\mathbb{R}}a(s,t)\phi(s) ds\right)^q$ and
$q=\frac{p}{p-1}$.

He claims that $$ \int_{0}^{\infty}e^{-F(t)}dt = \int_{\infty}^{\infty}|E_{\lambda}|e^{-\lambda}d\lambda $$ where $E_{\lambda}:=\{t\geq 0:F(t)\leq\lambda\}$. I don't see how to prove it. I was thinking of the Layer cake representation but I couldn't prove it as the integral in R.H.S is over $\mathbb{R}$ not in $(0,\infty)$.

Furthermore he claims that if $t\in E_{\lambda}$ then $$ t-\lambda\leq((b^q+t)^{\frac{1}{q}}(1-L(t)^p)^{\frac{1}{p}}+bL(t))^q $$ where $L(t)=\left(\int_s^{\infty}\phi(s)^p ds\right)^{\frac{1}{p}}$ (here the 1st term of RHS comes out from using Holder inequality and equations \eqref{1}, \eqref{2}, but the last portion i.e $+bL(t)$ is not coming rather some weaker estimate is coming.

Any help would be very much helpful to understand the thing going on.