g is a continous function on the interval $[0,1]$, and we have
$$lim_{n \to \infty} \int_0^1...\int_0^1 g(\sqrt[{n}]{z_1z_2...z_n})dz_1dz_2...dz_n$$
Prove whether the limit exists.
Would this be related to strong laws as it is not summation? I am confused since I do not know what g is in the first place so how would i know how to calculate g given a variable. Any hint about what lemma/law i should use would be appreiciated.
For $X_1,X_2,\ldots,X_n$ independent and uniformly distributed on $[0,1]$, $$ \int_0^1...\int_0^1 g(\sqrt[{n}]{z_1z_2 \dots z_n})dz_1dz_2...dz_n = \mathbb E\left[g\left(\sqrt[n]{X_1X_2\ldots X_n}\right)\right]=\mathbb E\left[g\left(\exp\left\{\frac{\ln X_1+\ln X_2+\ldots+\ln X_n}{n}\right\}\right)\right] $$ Then apply LLN and use continuity and boundedness of composition of two functions $g(x)$ and $\exp(x)$ to prove convergence of expectations.