\u8fd9\u6837, \u5f00\u5934\u7684\\par\u4f1a\u5728\u6bcf\u4e2astep\u7684\u7ed3\u5c3e\u81ea\u52a8\u6362\u884c.\u60c5\u770b\u4e0b\u9762\u7684\u4f8b\u5b50:<\/p>\n
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\n\\begin{proof}\nI would like to share my answer, it is a litter long, and I will just guide the proof:\n\\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 $R^d$;\n\\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 $R^d$;\n\\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 $\\phi_{k,j}\\chi_{Q_k}$)\n\\step Apply Egorov's Thm on $Q_k$ to select $J(k)$, such that $$|\\psi_k-\\phi_{k,J(k)}|<\\frac{1}{k},\\quad\\forall x\\in E_k\\bigcup Q_k^c.$$\nwhere $m(Q_k-E_k)<\\frac{1}{2^k}$.\n\\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 $E^c$.\n\\end{proof}\n<\/pre>\n\u6548\u679c\u5982\u56fe:
\n![\"\"](\"http:\/\/i.imgur.com\/ibxJmsr.png\" "\\"\\"")
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\u8fd9\u65f6, Step \u6240\u5728\u6bb5\u843d\u4f1a\u81ea\u52a8\u7f29\u8fdb.<\/p>\n","protected":false},"excerpt":{"rendered":"
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