Uniform deviation bound for a finite unit-range class

Given: IID observations, a finite nonempty measurable $[0,1]$-valued class, $n\ge 1$, and $ϵ>0$.

$P_{n}f=n^{-1}{\sum }_{i}f(Z_i)$ and $Pf=\int fdP$ (Empirical, centered, and Rademacher processes).

IID means independent and identically distributed (Identical distribution and IID families); independence of random elements is defined through their generated sigma-algebras (Independent random elements).

Measurable functions of independent random elements remain independent (Measurable coordinatewise functions preserve independence).

If $Z$ has law $P$, then $\mathbb{E}f(Z)=\int fdP$ for integrable $f$ (Law or distribution of a random element, Change of variables for expectation).

If $0\le Y\le 1$ almost surely, then $\mathbb{E}{e}^{\lambda (Y-\mathbb{E}Y)}\le e^{{\lambda }^2\mathrm{/}8}$ for all real $\lambda$ (The Hoeffding MGF bound for a variable in a unit interval).

For independent real random variables $X_i$ and Borel maps $g_i$, the expectation of a finite product of the nonnegative variables $g_i(X_i)$ factors into the product of expectations; also, $\exp (x+y)=\exp (x)\exp (y)$ (Expectations factor over finite products of independent random variables, The exponential addition formula $\exp (x+y)=\exp (x)\exp (y)$).

For nonnegative $W$ and $a>0$, $P(W\ge a)\le \mathbb{E}W\mathrm{/}a$ (Markov's inequality for random variables).

A finite union of events has probability at most the sum of their probabilities (Basic identities for a probability measure).

For $c\in \mathbb{R}$, the translation $t\mapsto t-c$ is Borel measurable because its strict upper threshold set at $a$ is $(a+c,\mathrm{\infty })$. For $\lambda \in \mathbb{R}$, the map $t\mapsto e^{\lambda t}$ is Borel measurable: its upper threshold set is $\mathbb{R}$ for $a\le 0$; for $a>0$ it is $\mathbb{R}$ if $\lambda =0$ and $a<1$, empty if $\lambda =0$ and $a\ge 1$, $(\log (a)\mathrm{/}\lambda ,\mathrm{\infty })$ if $\lambda >0$, and $(-\mathrm{\infty },\log (a)\mathrm{/}\lambda )$ if $\lambda <0$. These formulas use $\exp (0)=1$, positivity, strict increase, and the inverse relation of $\exp$ and $\log$. Borel maps preserve measurability under composition, and finite arithmetic and lattice operations preserve measurability (The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$, The natural logarithm as the inverse of the exponential function, The real exponential function and the number $e$ by a power series, Composition with a Borel measurable outer map preserves measurability, The exponential function is strictly increasing, Threshold characterisations of real-valued and extended-real-valued measurability, Arithmetic and lattice operations preserve measurability whenever they are defined).