In most discussions of concentration inequalities or calculations of rejection region in hypothesis testing, the measure used is left vague. For example, for independent random variables $X_1, \ldots, X_n$ satisfying $0 \le X_i \le 1$, Hoeffding's inequality is usually stated (Wikipedia link) as
$$\text{P}\left(\overline{X} - \mathbb{E} \overline{X} \ge t\right) \le e^{-2nt^2} \tag{1}$$
where $\overline{X} = \frac{1}{n}(X_1 + \cdots + X_n)$ and $t \ge 0$.
To make this concrete, suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is the underlying probability space. Then do we treat $\overline{X}$ as a function on $\Omega$ or on $\Omega^n$, because that determines if "$P$" in $(1)$ is $\mathbb{P}$ or $\mathbb{P}^{\otimes n}$ (product measure on $(\Omega^n, \mathcal{F}^{\otimes n})$. I think $\overline{X}$ should be a function on $\Omega$ looking at the proof of the inequality.
But now if you consider the general McDiarmid's inequality which states (or check Ledoux's book The Concentration of Measure Phenomenon): Let $(K, \mathcal{A}, \mu)$ be a probability space, let $L > 0$ be a constant, and let $f \colon K^n \to \mathbb{R}$ be a measurable function (for the product $\sigma-$algebra $\mathcal{A}^{\otimes n}$) which is $L-$Lipschitz for the normalized Hamming metric (i.e., $d(x,y) = \frac{1}{n}\left|\{i=1,\ldots,n : x_i \neq y_i\}\right|$ for $x,y \in \Omega^n$), then
$$\mu^{\otimes n} \left\{ f - \int_{K^n} f \,\mathrm{d}\mu^{\otimes n}\ge t \right\} \le e^{-2nt^2/L^2}$$
Let $K = [0,1]$, let $X_1, \ldots, X_n$ be i.i.d random variables (mapping $\Omega$ to $K$) with distribution $\mu$ (i.e., $\mu = \mathbb{P} \circ X_i^{-1}$), and let $f(x_1, \ldots, x_n) = \frac{1}{n}(x_1 + \cdots + x_n)$. Then clearly $f$ is $1-$Lipschitz. Denote by $X$ the function $(X_1, \ldots, X_n) \colon \Omega^n \to K^n$. Change of variables implies
$$\int_{K^n} f \, \mathrm{d}\mu^{\otimes n} = \int_{\Omega^n} f(X) \, \mathrm{d} \mathbb{P}^{\otimes n} = \int_\Omega X_1 \, \mathrm{d}\mathbb{P} = \mathbb{E}(X_1)$$ and we can write $$ f(X) = \overline{X} $$
Therefore, change of variables again and McDiarmid's inequality implies
$$\mathbb{P}^{\otimes n} \left\{ \overline{X} - \mathbb{E}(X_1) \ge t \right\} \le e^{-2nt^2} \tag{2}$$
And now compare equations $(1)$ and $(2)$. Why the discrepancy in measure?
