Layer cake representation

Remarks

The name comes from the pointwise identity $f(x) = \int_0^\infty \mathbf{1}_{\set{f > t}}(x) \, dt$. It writes $f$ as a stack of thin horizontal layers, one for each height $t$. Integrating over $X$ and exchanging the two integrals by Tonelli's theorem gives the formula. The second formula follows in the same way from $f(x)^p = \int_0^{f(x)} p t^{p-1} \, dt$.

Example: for $f(x) = x$ on $[0, 1]$ with Lebesgue measure, $d_f(t) = 1 - t$ for $0 \le t \le 1$ and $0$ after. Both sides equal $1/2$.

For a non-negative random variable $Y$ the first formula reads $\mathbb{E}[Y] = \int_0^\infty P(Y > t) \, dt$.

Going deeper

For a student

The formula turns integrals of $f$ into integrals of its distribution function. This is why the $L^p$ norms depend only on $d_f$, so equimeasurable functions and decreasing rearrangements have the same norms. It is the basic tool behind weak $L^p$ estimates, the Marcinkiewicz interpolation theorem and the bounds for the Hardy–Littlewood maximal function.