Question

A function : (0,0) - 0.00) is called a Young function, if 1. w is not identically 0. 2. lm, (u) = (0) = 0, 3. is convex on (0..) and lim, (u) = (b) (convention here poo) = o), where by = supu > 0: p(x) < oo). For a Young function we define = supu > 0: (u) = 0) du = sup{u € 0.6-): v(u/2) = v(u)/2) We also define Op(u) > 0:{u) + (0) = w), > 0. The complementary function : 0.00) - R of is defined by "(t) = sup(uv - plu): > 0) > 0. The following Young's Inequality holds true ut (u) + '() for all 0. Exp Gil Letp: 0,0) + 0,00] be a positive increasing function (that is p(t) > 0 for all 1 € (0.00) and ph p ta) whenever i 12. i 0,0)). We define generalized inverses g. 10, 0) (0.00 of p by 9 (8) = sup{t > 0:pu) s}, 4 (s) = sup[t> 0p(t) <s), (Note: we allow all the above functions to take value oc). ues (5 ) Let ( 1 ) be a (complete?) measure space and f:X (-0, be a measurable function which is finite pa.. on X. Lety: 0,00) - 10.o be a lower semi- continuous function. Prove or disprove that the composition (sl) is measurable this ERENCES en u

Answer 1

SOLUTION:

Given data from above question $\varphi$ is not identically '0'

Then

The second and third condition states that the function $\varphi$ is convex and is lower semi-continuous. By the Fenchel-Moreau theorem,

we know that for any extended real valued function in a Hausdorff locally convex space,

  $f^{**}=f$

if and only if is convex and is lower semi-continuous (or identically $-\infty$ or $\infty$ ). Since $\varphi$ satisfies the conditions.