巧用对称化三重积分为三次

$f(x)$ 在 $0\leq x\leq1$上连续。
证明:
$\int _0^1\ dx\int _x^1\ dy\int _x^y\ f\left(x\right)f\left(y\right)f\left(z\right)dz=\frac{1}{3!}\left(\int _0^1\ f\left(t\right)dt\right)^3$

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证明如下：

$\left(\int_0^1 f(x)>dx\right)^3=6\int_{0\leq x\leq z\leq y\leq1}f(x)f(y)f(z)d(x,y,z)=6\int_0^1\int_x^1\int_x^y f(x)f(y)f(z)dzdydx$