Show that $\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$

Question

Let $g:[0,1]\mapsto\mathbb{R}$ be a continuous function, and $\lim_{x\to0^+}g(x)/x$ exists and is finite. Prove that $\forall f:[0,1]\mapsto\mathbb{R}$,

$$\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$$

Answer 1

I'm a little late to the game and see that this suggestion has been made, but let's carry it out.

Let $y=x^n$, $x=y^{1/n}$, $dy = (1/n) y^{1/n} dy/y$. Then

$$n \int_0^1 dx \: f(x) g(x^n) = \int_0^1 \frac{dy}{y} y^{1/n} f(y^{1/n}) g(y)$$

As $n \rightarrow \infty$, $y^{1/n} \rightarrow 1 \forall y \in (0,1]$; that is, apart from $y=0$; however, since this is an isolated point (i.e., measure zero), then we have

$$\lim_{n \rightarrow \infty} n \int_0^1 dx \: f(x) g(x^n) = f(1) \int_0^1 dy \frac{g(y)}{y}$$

as was to be shown.