The task is to use SLLN to find limits:
a. $\lim_{n \to \infty} \int_{0}^{1} ... \int_{0}^{1} \frac{x_1^3 + ... + x_n^3}{x_1 + ... + x_n} dx_1 ... dx_n$
b. $\lim_{n \to \infty} \int_{0}^{1} ... \int_{0}^{1} f(\sqrt[n]{x_1 ... x_n}) dx_1 ... dx_n$ where $f : [0,1] \to R$ is continuous.
What I've tried: If $a_n$ is a series of sample means size $n$ from population with mean $\mu$ then $a_n \to \mu$, so I tried to write integrands as some functions g(x) of sample mean. Then limits would be integrals of $g(\mu)$. Then I realised that 1. I don't know how to rewrite integrands as functions of sample mean 2. It doesn't make sense because I don't know anything about the distribution the sample comes from and I don't know $\mu$.
Can anyone hint me on how to approach this task?
Hints: Let $X_1,X_2,...$ be an i.i.d sequence with uniform distribution on $(0,1)$. Then the first limit is $\lim_{n \to \infty} E\frac {X_1^{3}+X_2^{3}+\cdots +X_n^{3}} {X_1+X_2+\cdots+X_n}$. Divide numerator and denominator in the ratio by $n$. By SLLN the limit is $\frac {EX_1^{3}} {EX_1}$. [The ratio lies between $0$ and $1$ so DCT can be applied].
For the second part let $g(x)=f(e^{x})$. Then the given limit is $\lim Eg(\frac {\ln X_1+\ln X_2+\cdots+ln X} n)=g(E \ln X_1)$.