1
$\begingroup$

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 ?

$\endgroup$
0
$\begingroup$

$\times$ is the Cartesian product. It means the function takes two inputs, one being an element of $X$ and one being an element of $\Omega$.

$\endgroup$
0
$\begingroup$

It's the Cartesian Product, which will give you a set of ordered pairs. Using your $X$ and $\Omega$, it's defined as such:

$$X = \{X_1, X_2, ~...~ , X_n\}\\ \Omega = \{\Omega _1, \Omega_3, ~...~ , \Omega_n \}$$ $$ X \times \Omega = \{(X_1, \Omega_1), (X_1, \Omega_2), ~...~, (X_1, \Omega_n), (X_2, \Omega_1), (X_2, \Omega_2), ~...~, (X_n, \Omega_n) \}$$

An easier-to-read example might be something like:

$$A = \{1, 2\}\\ B = \{3, 4\}\\ A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$$

It's a pretty simple operation that, basically, gives you all combinations of elements of both sets in the order in which they appear in the Cartesian product.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.