Suppose the random vector $X = (X_1,...,X_n)$ has a distribution on $R^n$ which has a density $f$ with respect to Lebesgue measure on Rn and that $f$ is of the form $f(x_1, x_2,...,x_n) = g_1(x_1)g_2(x_2)··· g_n(x_n)$ where the $g_i$ are positive and measurable. Show that X_1,...,X_n are independent.
I know if $X$ and $Y$ are independent random variables, and $g(·)$ and $h(·)$ are measurable and real valued then $g(X)$ and $h(Y )$ are independent. how can I use this result to my current problem. Or any other hints would be really appreciated
If $r_i:=\int g_i(y)dy$ then we have $r_i\geq0$ and moreover: $$1=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} f(x_1,\dots,x_n)dx_1\dots dx_n=r_1\times\cdots\times r_2$$
Then we conclude that $r_i>0$ for each $i\in\{1,\dots,n\}$.
Taking $f_i(x):=\frac{g_i(x)}{r_i}$ we have $f(x_1,\dots,x_n)=f_1(x_1)\cdots f_n(x_n)$ but this with $\int f_i(y)dy=1$ for each $i\in\{1,\dots,n\}$.
It can be shown now that $f_i$ serves as PDF of $X_i$.
As an example let's do it for $i=1$.
$$F_{X_1}(x)=\int_{-\infty}^x\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} f_1(y_1)f_2(y_2)\cdots f(y_n)dy_1dy_2\cdots dy_n=$$$$\int_{-\infty}^xf_1(y_1)dy_1\int_{-\infty}^{\infty}f_2(y_2)dy_2\cdots\int_{-\infty}^{\infty}f_n(y_n)dy_n=\int_{-\infty}^xf_1(y_1)dy_1$$
Then based on these results it can be shown that $$F_{X_1,\dots,X_n}(x_1,\dots,x_n)=F_{X_1}(x_1)\times\cdots\times F_{X_n}(x_n)$$
revealing that the $X_i$ are independent.