Let $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ be a measure space. Let $g:\mathbb{R}\rightarrow \mathbb{R}$, be a continous function with
$$g\_n(x)=g(x^n)\mathbb{1}\_{[0,1]}(x), \qquad x\in \mathbb{R}, n\in \mathbb{N}.$$
I need to calculate
$$\lim\_{n\rightarrow \infty} \int g\_n d\lambda$$
And I am not sure if I have done this correct, since it depends on the function $g$, anyways, I have done the following:
\begin{align}
\lim\_{n\rightarrow \infty} \int g\_n d\lambda &=\lim\_{n\rightarrow \infty} \int g(x^n)\cdot \mathbb{1}\_{[0,1]}(x)\lambda(dx)\\
&=\lim\_{n\rightarrow \infty} R\int\_0 ^1g(x^n)dx \qquad (g:continous)\\
&=\lim\_{n\rightarrow \infty}\left[g(x^n)x\right]\_0^1=g(1).
\end{align}
Any help is much appreciated.