How is this colored background text achieved?

Question

I saw the following beautifully colored text for definitions, theorems, examples... in someone's notes. How do we achieve this in latex?

Answer 1

Here is a way to reproduce the second image with ntheorem, mdframed and xcolor:

\documentclass[11pt]{book} %
\usepackage[T1]{fontenc}
\usepackage[svgnames]{xcolor}%
\usepackage{amssymb,amsmath}
%\usepackage{framed}
\usepackage[amsmath,framed]{ntheorem}
\usepackage[ntheorem]{mdframed}
\theoremheaderfont{\sffamily\bfseries}
\theorembodyfont{\normalfont\upshape}
   \theoremstyle{plain}
\newmdtheoremenv[topline=false, bottomline=false, rightline=false, leftline=true, linewidth=2pt, linecolor=Purple, backgroundcolor=Thistle! 20, innertopmargin=10pt, innerbottommargin=10pt]{example}{Example}
\begin{document}
\begin{example}
The rule $1$ cannot follow. $1$ corresponds to
    [ A = \begin{bmatrix}
      1 & 1 \ 1 & 0
    \end{bmatrix} ]%
\end{example}
\end{document}

Answer 2

With tcolorbox

\documentclass{article}
\usepackage{tcolorbox, xcolor}
\usepackage{lipsum}
\begin{document}
\begin{tcolorbox}[colback=blue!15, sharp corners, boxrule=0pt, leftrule=1mm, colframe=blue]
\lipsum[1]
\end{tcolorbox}
\end{document}