TITLE: Compute the following limit $ I=\lim_n\int_{[0,1]^n}f(x)\,dx_1\ldots dx_n$. $f$ being the max function over $n$ coordinates.
QUESTION [5 upvotes]: Compute the following limit
$$
I=\lim_n\int_{[0,1]^n}f(x)\,dx_1\ldots dx_n,
$$
where 
$$
f:\mathbb{R}^n \to \mathbb{R},\ x=(x_1,\ldots,x_n) \mapsto \max\{x_1,\ldots,x_n\}.
$$

REPLY [3 votes]: Here is a marginally different approach:
Let $N=\{ x \in [0,1]^n | x_i \neq x_j, \ \forall I \neq j \}$. It should be clear that $m N = m [0,1]^n$, where $m$ is the Lebesgue measure. Let $T = \{ x \in N | x_i > x_{i+1},\  \forall i \}$, and note that $N = \cup_{\pi \in \Pi_n } \pi(T)$, where $\Pi_n$ is the set of functions that permute the variables in $[0,1]^n$. Furthermore, note that $\max(x_1,...,x_n) = \max([\pi(x)]_1,...,[\pi(x)]_n)$ for all $\pi \in \Pi_n$.
Hence $I_n = n! \int_T x_1 dx = \int_{x_1=0}^1\int_{x_2=0}^{x_1} \cdots \int_{x_n=0}^{x_{n-1}} x_1 dx = n! \frac{1}{(n+1)(n-1)!} = \frac{n}{n+1}$, and $\lim_n I_n = 1$.