consider the union bound, when H consists of only h1 and h2,

Pr[|Ein(g)-Eout(g)|>e]

<=Pr[|Ein(h1)-Eout(h1)|>e or |Ein(h2)-Eout(h2)|>e] （eq.1）

<=Pr[|Ein(h1)-Eout(h1)|>e] + Pr[|Ein(h2)-Eout(h2)|>e] (eq.2）

=2exp(-2Ne^2) (eq.3)

i wonder which kind of h1 and h2 can more "accurately" satisfy the union bound, maybe this means union bound equals vc bound.

in other words, can i find h1 and h2 to let (eq.1)=(eq.2)?

Pr[|Ein(h1)-Eout(h1)|>e or |Ein(h2)-Eout(h2)|>e]

=Pr[|Ein(h1)-Eout(h1)|>e and |Ein(h2)-Eout(h2)|>e] (eq.a)

+Pr[|Ein(h1)-Eout(h1)|>e and |Ein(h2)-Eout(h2)|<=e] (eq.b)

+Pr[|Ein(h1)-Eout(h1)|<=e and |Ein(h2)-Eout(h2)|>e] (eq.c)

if (eq.2)=(eq.1), then,

sup(eq.a)=0

sup(eq.b)=sup(eq.c)=1/2(eq.3)=exp(-2Ne^2)

but i cannot find the example h1&h2, which means h1 and h2 has no overlap when considering the union bound.

can u help?

thanks.