This code:
\documentclass[twoside,symmetric,notoc,justified]{tufte-book}%
\usepackage[utf8]{inputenc}
\usepackage[latin]{babel}
\usepackage[OT1]{fontenc}
\usepackage{lipsum}
\geometry{
%showframe,
paperwidth=145mm,
paperheight=215mm,
inner=16mm,
outer=10mm,
top=10mm,
bottom=40mm,
marginparsep=3mm,
marginparwidth=40mm,
includemp,
includehead,
}
\fancyhf{}%
\fancyhead[LE]{\bfseries\thepage\quad$\bigg|$\quad\leftmark}
\fancyhead[RO]{\bfseries\rightmark\quad$\bigg|$\quad\thepage}
\begin{document}
\tableofcontents
\chapter*{Introduction}
\addcontentsline{toc}{section}{\textbf{Introduction}}
\lipsum[1-5]
\chapter{Chapter1}
\section{Section1}
\lipsum
\section{Section2}
\lipsum
\section{Section3}
\lipsum
\chapter{Chapter2}
\section{Section4}
\lipsum
\section{Section5}
\lipsum
\end{document}
As you see, the headings still show "Index", but it is already "Introduction". How to fix it? I want it to show "Introduction" on the page where it is. Thank you.

\chapter*. You can either remove the*or write\chapter*{Introduction} \markboth{Introduction}{Introduction}. – Jan 25 '18 at 21:12\addcontentslineif you remove the asterisk.\chapter*suppressing adding the chapter to the TOC. – godbyk Jan 25 '18 at 21:20