如何写证明中的Step1, Step2等等

Vanabel/ 4月 16, 2013/ LaTeX/ 0 comments

\newcounter{stepnum}
\newcommand{\step}{%
\par
\refstepcounter{stepnum}%
\textbf{Step \arabic{stepnum}}.\enspace\ignorespaces
}

证明 . I would like to share my answer, it is a litter long, and I will just guide the proof:
\step For $f=\chi_E$, with $m(E)< \infty$, see Stein's Book Thm4.3: there exist step function $\phi_k\to\chi_E$ a.e. in $R^d$;
\step For general simple function $\psi=\sum_{l=1}^n a_l\chi_{E_{l}}(x)$ (note that by definition $m(E_l)<\infty$, $\forall l$), there exists step function $\phi_{k}$, such that $\phi_{k}\to\psi$ a.e. in $R^d$;
\step By Stein's book Thm4.2, for measurable function $f$ on $R^d$, there exists simple function $\psi_k\to f$ point-wise in $R^d$. Thus, by step 2. there exist step function $\phi_{k,j}\to\psi_k$, $\forall k$. We can select a large enough cubic $Q_k$, such that $phi_{k,j}\equiv \psi_k$ in $Q_k^c$ and $\phi_{k,j}\to\psi_k$. (Replace $\phi_{k,j}$ by $\phi_{k,j}\chi_{Q_k}$)
\step Apply Egorov's Thm on $Q_k$ to select $J(k)$, such that $$|\psi_k-\phi_{k,J(k)}|<\frac{1}{k},\quad\forall x\in E_k\bigcup Q_k^c.$$
where $m(Q_k-E_k)<\frac{1}{2^k}$.
\setp Let $E=\limsup_{k=1}^\infty (Q_k-E_k)$, prove that $m(E)=0$, and $\phi_{k,J(k)}\to f$ a.e. in $E^c$.