My textbook said:
Let $E\subset\mathbb{R}^n$, let $G$ be an open set, and let $|\cdot|_e$ denote outer measure.
if $\exists{}G$ s.t. $E\subset{}G$ and $|G-E|_e\lt\varepsilon$ for an any given $\varepsilon$, then $E$ is measurable.
if $\{\mathbf{x}\in{}E: f(\mathbf{x})\gt{}a\}$, for every finite $a$, is measurable, then $f$ is measurable.
if $\int_E f$ exists and is finite, then $f$ is integrable on $E$.
Q1) My book common wrote that "~~~ said to be measurable if ~~~." Are converses of them correct? that is,
$\exists{}G$ s.t. $E\subset{}G$ and $|G-E|_e\lt\varepsilon$ for an any given $\varepsilon$, if $E$ is measurable.
$\{\mathbf{x}\in{}E: f(\mathbf{x})\gt{}a\}$, for every finite $a$, is measurable, if $f$ is measurable.
$\int_E f$ exists and is finite, if $f$ is integrable on $E$.
Q2) is "for every finite $a$" equal to writing "for any $a\in\mathbb{R}$"?
Q3) Is the following theorem correct?
If $f$ is measurable, then $f$ is integrable over $E\subset\mathbb{R}^n$.
If $f$ is integrable over $E\subset\mathbb{R}^n$, then $f$ is measurable.