Lou Gehrig died on this day in 1941 of amyotrophic lateral sclerosis (ALS) which is commonly called Lou Gehrig's Disease. He was 37 years old.
Also on this day, Adolf Hitler and Mussolini had a length meeting at the Brenner Pass. Hitler ranted about Rudolph Hess but didn't see fit to tell Mussolini that a German invasion of the Soviet Union was imminent. Mussolini knew, however, due to troop movements bordering Italy that were impossible to conceal, something that should have also clued in Stalin, who seemed blinded to the reality of what was about to occur.
After the meeting Mussolini told an aid "I wouldn't be at all sorry if Germany in her war with Russia got her feathers plucked."
Ultimately, of course, Italy would end up contributing troops to the German effort.
German paratroops executed over 20 Cretan civilians as a reprisal for Cretan civilian resistance to the invasion of the island. They also photographed the event.
In France, the Vichy government ordered a census of the country's Jewish residents and took additional anti Jewish measures, which is noted here:
Today in World War II History—June 2, 1941
On the same day, France gave Germany port rights in Bizerte in Tunisia for the support of its army in North Africa, but restricted it to non military supplies.
