Intuition of $\int_{-\infty}^{\infty}$?

Question

What's the intuition of the improper integral

$$\int_{-\infty}^{\infty}$$

Is it really integral over the entire domain $\mathbb{R}$?

Answer 1

$$\int_{-\infty}^{\infty} f(x) dx$$

Pick any $a \in \mathbb R$.

$$= \int_{-\infty}^{a} f(x) dx + \int_{a}^{\infty} f(x) dx$$

$$= \lim_{b \to -\infty} \int_{b}^{a} f(x) dx + \lim_{c \to \infty} \int_{a}^{c} f(x) dx$$

$$= \lim_{b \to -\infty} [F(a) - F(b)] + \lim_{c \to \infty} [F(c) - F(a)]$$

$$= \lim_{b \to -\infty} [- F(b)] + \lim_{c \to \infty} [F(c)]$$

We can see that

$$\int_{-\infty}^{\infty} f(x) dx \ \text{exists if and only if}$$

$$\lim_{b \to -\infty} [- F(b)] \ \text{and} \ \lim_{c \to \infty} [F(c)] \ \text{exist}$$