I am reading the book STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS wrote by prof.Mao. In page 6, there is a dominated convergence theorem in d dimension in $L^p$, which seems different from the other dominated convergence theorem I read in other books. It states like this:
dominated convergence theorem in d dimension in $L^p$
I am confused what is the "$|X_k|$" here mean? Is it a Euclidean norm operator? Because this transform a random variable in $R^d$ to $R$. What is it exact format? Some people told me that after $|\cdot|$ operator, it becomes a real number? Why? Is he true?
I think the condition they are stating is: $\mu(\{\omega \in \Omega: ||X_k(\omega)||_1 \leq Y(\omega)\}) = 1$
where $||.||_1$ is $L_1$ norm. Now we can directly apply the $1$-dimensional dominated convergence theorem since every component of $|X_k|$ is now bounded by $|Y|$ and the premise of 1-dimensional dominated convergence theorem holds. We can now apply limit to each component of $X_k$ and conclude the theorem.
The above is true. You can actually put other norms also $||X_k(\omega)||_{\ell}$ also works since all norms are equivalent in finite dimensional case $d < \infty$ i.e., $||X_k(\omega)||_1 \leq C(d) \times ||X_k(\omega)||_{\ell}$.
So the statement is true for all $p$ norms.