Question Number 75079 by ~blr237~ last updated on 07/Dec/19
![Let f∈C([0,1],[0,1]) Prove that lim_(n→∞) ∫_([0,1]^n ) f((1/n)Σ_(i=1) ^n x_i )dx_1 ....dx_n =f((1/2))](Q75079.png)
$$\mathrm{Let}\:\mathrm{f}\in\mathrm{C}\left(\left[\mathrm{0},\mathrm{1}\right],\left[\mathrm{0},\mathrm{1}\right]\right)\:\: \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{n}} } \mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{x}_{\mathrm{i}} \:\right)\mathrm{dx}_{\mathrm{1}} ....\mathrm{dx}_{\mathrm{n}} \:=\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$