Meaning of this cross product

Let $X$ be an open subset of $\mathbb{R}^{n}$ and $\Omega$ be a measure space. Define a function $f:X\times\Omega \longrightarrow \mathbb{R} .$

What is the meaning of $X\times\Omega$ ? I mean, is n-tuple column vector X multiplied by n-tuple row vector of $\Omega$ produces the real number? Please clear my confusion. How the $\Omega$ should look like ?

$\times$ is the Cartesian product. It means the function takes two inputs, one being an element of $X$ and one being an element of $\Omega$.
It's the Cartesian Product, which will give you a set of ordered pairs. Using your $X$ and $\Omega$, it's defined as such:
$$X = \{X_1, X_2, ~...~ , X_n\}\\ \Omega = \{\Omega _1, \Omega_3, ~...~ , \Omega_n \}$$ $$X \times \Omega = \{(X_1, \Omega_1), (X_1, \Omega_2), ~...~, (X_1, \Omega_n), (X_2, \Omega_1), (X_2, \Omega_2), ~...~, (X_n, \Omega_n) \}$$
$$A = \{1, 2\}\\ B = \{3, 4\}\\ A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$$