Suppose that $f, g$ are continuous on $[a,b]$. Show that if $f=g$ a.e. on $[a,b]$, then in fact $f=g$ on $[a,b]$.

Asked Oct 22 '21 at 11:11

Active Oct 22 '21 at 11:11

Viewed 64 times

Suppose that $f, g$ are continuous on $[a,b]$. Show that if $f=g$ a.e. on $[a,b]$, then in fact $f=g$ on $[a,b]$.

If $f=g$ a.e. on $[a,b]$, then $m(\{x \in [a,b] \mid f(x)\ne g(x)\}) = 0$. How can we conclude the fact that $f=g$ on $[a,b]$? I have that $\{x \in [a,b] \mid f(x)\ne g(x)\} \subset [a,b]$ so $$m(\{x \in [a,b] \mid f(x)\ne g(x)\}) =0\le m([a,b])= b-a$$ but no help from this.

asked Oct 22 '21 at 11:11

Christie

Hint: $f-g=0$ a.e., and use continuity. – fwd Oct 22 '21 at 11:23
The complement of a set of measure $0$ is dense. – Kavi Rama Murthy Oct 22 '21 at 11:26
Does this answer your question? functions $f=g$ $\lambda$-a.e. for continuous real-valued functions are then $f=g$ everywhere – zwim Oct 22 '21 at 12:59

Suppose that $f, g$ are continuous on $[a,b]$. Show that if $f=g$ a.e. on $[a,b]$, then in fact $f=g$ on $[a,b]$.

0 Answers0