Why Lebesgue measure? Why Borel σ-algebra?

1. Is any measure on any σ-algebra inside the power set of $\mathbb{R}^d$ a formal definition (or generalisation) of "volume" in $\mathbb{R}^d$?

2. What's so special about Lebesgue measure that we choose it as the standard way to assign measure to subsets of $\mathbb{R}^d$?

3. What's so special about Borel σ algebra? Why not other σ-algebra?

4. Is there a measure on the Borel σ algebra of $\mathbb{R}^d$ such that $\gamma ((a,b])$ may not be $b-a$?

For question 2, I guess Lebesgue measure is chosen as the standard way because it's the unique measure on the Borel σ algebra of $\mathbb{R}^d$ such that $\gamma ((a,b])=b-a$.

But I'm not sure if that's the reason, I'm not even sure if the important bit is the "Borel σ algebra" or "$\gamma ((a,b])=b-a$".

Any help will be appreciated!