If random variable A is independent of random vector (B,C), is A independent of C?

Question

If R.V. A is independent of a random vector (B,C), is A necessarily independent of C?

Answer 1

By independence of a and vector (b,c), we have $$f(a,b,c)=f(a)f(b,c)$$ So, integrating out b, $$\int_{-\infty}^{\infty} f(a,b,c)\,db = \int_{-\infty}^{\infty} f(a)f(b,c)\,db \implies f(a,c)=f(a)f(c) $$

Answer 2

If you go back to the definition of independence, you can prove this statement without assuming the joint distribution of $(A,B,C)$ to be continuous.

$A$ and $(B,C)$ are independent if every $A$-measurable event is independent of every $(B,C)$-measurable event. Since every $C$-measurable event is $(B,C)$-measurable, conclude that every $A$-measurable event is independent of every $C$-measurable event. That is, $A$ and $C$ are independent.