Menu Close

Let-f-C-0-1-0-1-Prove-that-lim-n-0-1-n-f-1-n-i-1-n-x-i-dx-1-dx-n-f-1-2-




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))
$$\mathrm{Let}\:\mathrm{f}\in\mathrm{C}\left(\left[\mathrm{0},\mathrm{1}\right],\left[\mathrm{0},\mathrm{1}\right]\right)\:\: \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{n}} } \mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{x}_{\mathrm{i}} \:\right)\mathrm{dx}_{\mathrm{1}} ….\mathrm{dx}_{\mathrm{n}} \:=\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *