Can you give me an example of $A,B,C \subset{\mathbb{R}}$ with $A = B\setminus C$ but $\mu(A) \neq \mu(B) - \mu(C)$? [closed]

I'm given the measure of $$ \mu(\{x: f(x) > t\}) $$ for all $t$. And in solving a problem, I said that
$$ \mu(\{x: t_1 < f(x) < t_2\}) = \mu(\{x: f(x) > t_1 \}) - \mu(\{x: f(x) > t_2 \}) $$ But, looking back and thinking about it, I'm unsure if this is even true, despite it being intuitive.

Solution 1:

Let, $(X,{\scr{B}}(X),\mu)$ be a measure space.

$ B $ and $ C$ are measurable sets with $C\subset B$ and $\mu(C) <\infty$ , then your conclusion, $\mu(A) =\mu(B\setminus C) =\mu(B) - \mu(C) $ is true.

Because, \begin{align} \quad \mu(B) &=\mu(B \cap C) + \mu(B \cap C^c) \\ &= \mu(C) + \mu(B \setminus C) \end{align}

And, hence $\mu(B\setminus C) =\mu(B) - \mu(C) $$(\mu(C) <\infty ) $

Consider,$(\mathbb{R},{\scr{L}}(\mathbb{R}), m) $

$ (-\infty, 0) =\mathbb{R} \setminus [0, \infty) $

(all are Borel sets and so Lebesgue measurable)

And applying above property(excluding the possibility of finite measure of $C$)

$m\{(-\infty,0)\} =m\{\mathbb{R}\} - m\{[0,\infty ) \}$

Hence, $\infty=\infty -\infty$(!)

Hence, $A=B\setminus C$ doesn't imply $\mu(A) =\mu(B) -\mu(C) $