使用多个“\paragraph”时出错

Question

我正在尝试使用 \paragraph 连续创建两个段落：

\begin{proof}
\paragraph{Case 1}
If $ \mu E_N = \infty $ for some $ N $, then $ \mu E_m = \infty $
for all $ m \geq N $. Also $ \cup E_n \supseteq E_N $. So $ \mu (\cup E_n ) = \infty $.

\paragraph{Case 2}
If $ \mu E_n < \infty $ for all $ n $, then set $ C_1 := E_1, C_n := E_n\setminus E_{n-1} $. Then $ \cup E_n := \sqcup C_n $. By our lemma, $ \mu C_n = \mu E_n - \mu E_{n-1} $. Then 
    \begin{align*}
        \mu (\cup E_n) &= \mu (\sqcup C_n) \\
                       &= \sum_{n=1}^{\infty} \mu C_n \\
                       &= \lim_{N\to \infty } \sum_{n=1}^N \mu C_n \\
                       &= \lim_{N\to \infty } \mu E_N
    \end{align*}
\end{proof}

但是 pdflatex 给出错误：

! LaTeX Error: Something's wrong--perhaps a missing \item.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              

l.319   \paragraph
                 {Case 2}

我试过

\begin{enumerate}[label=Case(\arabic*)] ... \end{enumerate} 

如错误消息所示，但标签与其余文本不一致。

Answer 1

在第一个之前您需要一些文本\paragraph。

另一方面，我看不出它们有什么用处，所以我提供了不同的代码，修复了符号和墓碑。特别注意\bigcup。

\documentclass{article}
\usepackage{amsmath,amsthm}

\begin{document}

\begin{proof}
We divide the proof into two cases.

\paragraph{Case 1}
If $ \mu E_N = \infty $ for some $ N $, then $ \mu E_m = \infty $
for all $ m \geq N $. Also $ \cup E_n \supseteq E_N $. So $ \mu (\cup E_n ) = \infty $.

\paragraph{Case 2}
If $ \mu E_n < \infty $ for all $ n $, then set $ C_1 := E_1, C_n := E_n\setminus E_{n-1} $. 
Then $ \cup E_n := \sqcup C_n $. By our lemma, $ \mu C_n = \mu E_n - \mu E_{n-1} $. Then 
    \begin{align*}
        \mu (\cup E_n) &= \mu (\sqcup C_n) \\
                       &= \sum_{n=1}^{\infty} \mu C_n \\
                       &= \lim_{N\to \infty } \sum_{n=1}^N \mu C_n \\
                       &= \lim_{N\to \infty } \mu E_N
    \end{align*}
\end{proof}

\begin{proof}
We divide the proof into two cases.

\textbf{Case 1.}
If $ \mu E_N = \infty $ for some $ N $, then $ \mu E_m = \infty $
for all $ m \geq N $. Also $ \bigcup E_n \supseteq E_N $. So $ \mu (\bigcup E_n ) = \infty $.

\textbf{Case 2.}
If $ \mu E_n < \infty $ for all $ n $, then set $ C_1 := E_1, C_n := E_n\setminus E_{n-1} $. 
Then $ \bigcup E_n := \bigsqcup C_n $. By our lemma, $ \mu C_n = \mu E_n - \mu E_{n-1} $. Then 
    \begin{align*}
        \mu \Bigl(\bigcup E_n\Bigr) &= \mu \Bigl(\bigsqcup C_n\Bigr) \\
                       &= \sum_{n=1}^{\infty} \mu C_n \\
                       &= \lim_{N\to \infty } \sum_{n=1}^N \mu C_n \\
                       &= \lim_{N\to \infty } \mu E_N \qedhere
    \end{align*}
\end{proof}

\end{document}

Answer 2

要将标签与enumerate 左边距对齐，只需使用此代码：

\usepackage{enumitem}
......
\begin{enumerate}[label=Case(\arabic*), wide=0pt, font=\bfseries] 
\item ...
......... 
\end{enumerate}