If A, B are independent from C, then does it follow that B is independent from C

Question

Consider 3 random variables, A B and C. If A, B are independent from C, then does it follow that B is independent from C if so, how can we show this?

Answer 1

Yes.

Since you indicate they are random variables, I prefer to use the symbols of $X, Y$ and $Z$. That is if $X, Y$ are independent of $Z$, does it follow that $Y$ is independent of $Z$? The answer is YES. There are multiple ways to prove this rigorously. But fundamentally it is because if $\sigma(Z)$ is independent of $\sigma(X, Y)$, then $\sigma(Z)$ is independent of any sub-$\sigma$ algebra of $\sigma(X, Y)$, and $\sigma(Y) \subset \sigma(X, Y).$

Answer 2

For random variables, $A,B\perp C$ means that for all measureable subsets $\Sigma,\Delta,\Gamma$ of the variables' supports:

$$\mathsf P(A\in\Sigma, B\in\Delta,C\in\Gamma)=\mathsf P(A\in\Sigma, B\in\Delta)\cdot\mathsf P(C\in\Gamma)$$

Now as this is true for all subsets, ...