Need help with 2.2.1

[latidoremi]

The way I understand stratified sampling is this:

so naturally

but what is this fractional volume doing here?

and I don't understand these two at all, each stratum is an integral over lambda i, shouldn't the integrand be f(x) itself?

[zxdan-git]

I was also confused about this part. I asked Chatgpt and got an answer that makes sense to me. Chatgpt mentioned that the above equations are based on an assumption that each sample in a stratum is drawn uniformly and the fractional volume of stratum $$v_i = \int_{\Lambda_i}dx$$.

Therefore, if $$X_{ij}$$ is drawn uniformly in $$\Lambda_i$$, $$p(X_{ij}) = \frac{1}{v_i}$$.
$$E(f(X_{ij})) = \int_{\Lambda_i} f(x)p(x)dx = \frac{1}{v_i}\int_{\Lambda_i}f(x)dx$$

Not sure if it is the case.