Let me define the notions first,
(1)$\textbf{Young Function:}$ A convex function $\Phi:\mathbb{R}\to \mathbb{R^+}$ is said to be a Young function if the following conditions satisfy :
(a) $\Phi(0)=0$
(b)$\Phi(-x)=\Phi(x)$
(c) $\lim_{x\to\infty}\Phi(x)=+\infty$
(2) $\textbf{Orlicz Space :}$ Let $(\Omega,\Sigma,\mu)$ be a measurable space,then the set defined by
$L^\Phi(\mu)$={$f\colon \Omega \to \mathbb{R}$ such that $\int_{\Omega}\Phi(kf)d\mu<\infty$ for some $k>0$} is called an Orlicz space.
(3) $\textbf{Norm on an Orlicz Space:}$ we define the norm on the above Orlicz space as
$N_{\Phi}(f)$=inf {$k>0$ such that $\int_{\Omega}\Phi(\frac{f}{k})d\mu\leq1$}
Okay, so my questions are
(1) Why do we take the Convex function with some special properties as we defined in 1:(a),(b),(c)?
(2)Also we have defined the norm with $\int_{\Omega}\Phi(\frac{f}{k})d\mu\leq1$ so what will happen if this integral value is greater than one.
Please help, actually, I am trying to understand the definition of Orlicz space properly, your answer will help to understand it in a better way. Thank you
(1) When you try to prove $N_\Phi$ is a (semi)norm, I think you will see why these are needed. For example: For the triangle inequality, you will need $\Phi$ to be convex. (b) will let you prove $N_\Phi(f) = N_\Phi(-f)$. (a) will let you prove the constant $0$ is in $L^\Phi$, even if $\mu$ is an infinite measure; (a) is also used in (2), below. If (c) fails then $\Phi(x) = 0$ for all $x$, not very interesting.
(2) Show: $\Phi(x)$ is nondecreasing on $[0,+\infty)$. Note $\Phi(kf) = \Phi(k|f|)$. So if $\int \Phi(kf)\;d\mu < \infty$ for some $k$, use the dominated convergence theorem to show $\lim_{k \to 0^+} \int \Phi(k|f|)\;d\mu = 0$ using (a), so (taking $t=1/k$) there exists $t>0$ with $\int \Phi(\frac{|f|}{t})\;d\mu < 1$. In the definition of $N_\Phi(f)$, it is an infimum over a nonempty set of positive numbers.