The curly brace is too big

Question

\documentclass{article}
\begin{document}
\begin{equation*}
P^*(A) = \begin{cases}
1 & \quad \omega_0 \in A, \\
0 & \quad \omega_0 \notin A.
\end{cases}
\end{equation*}
\end{document}

When I used the above code to display a piecewise expression, I got the following:

Clearly, the curly brace appeared in the expression is too big, how can I adjust it so the display is more satisfactory? Thanks in advance.

P.S: My preamble:

\usepackage{setspace} 
\doublespacing
\usepackage[margin = 1.5in]{geometry}
\usepackage{float} 
\usepackage{graphicx}
\usepackage{color} 
\usepackage{amsmath} 
\usepackage{amsfonts} 
\usepackage{amsthm}
\usepackage{bm} 
\usepackage{framed}
\usepackage{verbatim}
\usepackage{natbib}
%\usepackage[symbol*]{footmisc}
\usepackage[stable]{footmisc}
\usepackage{mathtools}
\usepackage{mathrsfs}
\usepackage{amssymb}
\usepackage{hyperref}



\newcommand{\eps}{\varepsilon}
\newcommand{\rational}{\mathbb{Q}}
\newcommand{\real}{\mathbb{R}}
\newcommand{\integer}{\mathbb{Z}}
\newcommand{\nn}{\mathbb{N}}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\Ell}{\mathcal{L}}
\newcommand{\mean}{\mathbb{E}}
\newcommand{\prob}{\mathbb{P}}
\newcommand{\dd}{\mathop{}\!\mathrm{d}}
\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern1.5mu}\mkern 1.5mu}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\diam}{diam\,}
\newcommand{\Riemann}{\mathscr{R}}

\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\newtheorem{defns}{Definitions}[section]
\newtheorem{exmp}{Example}[section]

\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem*{cor}{Corollary}

\theoremstyle{remark}
\newtheorem*{rem}{Remark} 

Answer 1

The cases environment does \renewcommand{\arraystretch}{1.2}, which means it uses 20% more space than normal, but of course this is wrong when the document is already double spaced.

\documentclass{article}
\usepackage{amsmath}
\usepackage{setspace}
\usepackage{etoolbox}

\usepackage{lipsum}

\doublespacing

\makeatletter
\patchcmd{\env@cases}{1.2}{0.72}{}{}
\makeatother

\begin{document}

\lipsum*[2]
\begin{equation*}
P^*(A) = \begin{cases}
1 & \quad \omega_0 \in A, \\
0 & \quad \omega_0 \notin A.
\end{cases}
\end{equation*}
\lipsum[3]

\end{document}

Note that \doublespacing does \linestretch{1.667} and 0.72=1.2/1.667

Answer 2

Package amsmath is missing, which defines environments equation* and cases:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
P^*(A) = \begin{cases}
1 & \quad \omega_0 \in A, \\
0 & \quad \omega_0 \notin A.
\end{cases}
\end{equation*}
\end{document}

The brace looks normal sized.

Updated question

The preamble uses

\usepackage{setspace}
\doublespacing

As set, the increased line spacing also affects cases (and array, tabular). Workaround: A local switch to \singlespacing:

\documentclass{article}
\usepackage{setspace}
\doublespacing
\usepackage{amsmath}
\begin{document}
\begin{equation*}
P^*(A) = \begin{cases}
1 & \quad \omega_0 \in A, \\
0 & \quad \omega_0 \notin A.
\end{cases}
\end{equation*}
\begingroup
  \singlespacing
  \begin{equation*}
  P^*(A) = \begin{cases}
  1 & \quad \omega_0 \in A, \\
  0 & \quad \omega_0 \notin A.
  \end{cases}
  \end{equation*}
\endgroup
\end{document}

Answer 3

Compare this two cases

\documentclass{article}
    \usepackage{amsmath}

\begin{document}
\begin{equation*}
P^*(A) = \begin{cases}
1 & \quad \omega_0 \in A, \\[1ex]
0 & \quad \omega_0 \notin A.
\end{cases}
    \quad
P^*(A) = \begin{cases}
1 & \quad \omega_0 \in A, \\
0 & \quad \omega_0 \notin A.
\end{cases}
\end{equation*}
\end{document}

Both have equal big curly brace, but at the first one can has filling, that it is smaller or that it better fit content after it:

Difference between them is vertical space between equations: the first one has bigger. Maybe is this what you looking for.

Edit: Apparently the original question was misleading for me ... Problem caused by increased line spacing are well elaborate in Heiko Oberdiek and egreg answers, however, after considering any of proposed solutions, one still can manually increase interline spacing between equations (when this is appropriate) what make curly braces visually smaller.