The results of this Personality Test is quite surprisingly accurate, do give it a try to see if you are a Careerist, Entrepreneur, Harmonizer, Idealist, Hunter, Internationalist or Leader?
Benefits of doing the (Free) Career Test:
- Get familiar with the top companies in Singapore
- There are seven distinct career types based on career preferences, goals and personality. Get to know yours! (My result: Harmonizer)
- Take part in the annual Universum survey and win prizes!
Let be a finite measure space. Suppose that is a measurable function on . Let for each . Show that is integrable if and only if .
This proof has a cute solution that is potentially very short. We will elaborate more on this proof. Other approaches include using Markov’s Inequality / Chebyshev’s inequality.
Proof: Consider .
Note that for each on , , while . Therefore on .
Integrating with respect to , we get .
(=>) Now assuming f is integrable, i.e. , we have . . Therefore .
(<=) Conversely, if , then .
We are done.
Note: For a more rigorous proof of we can use MCT (Monotone Convergence Theorem).
Let . Then . By MCT, .