The German word meinetwegen is used in several different contexts. Its first meaning can be translated as "on my account" or "for me" in English.

BANNER PLACEHOLDER

Ich weiß, ihr seid nicht meinetwegen gekommen.

I know you haven't come because of me.

Caption 3, Verstehen Sie Spaß? - Sascha Grammel

 Play Caption

Hat Themba wirklich nur meinetwegen ein Eigentor gemacht?

Did Themba really score a goal against his own team just because of me?

Caption 33, Die Pfefferkörner - Eigentor - Part 6

 Play Caption

It can also mean "for example" or "assuming that":

Und wenn du dich jetzt meinetwegen dort auf die Seite drehst...

And if you now turn onto your side, for example...

Caption 63, Die Schmerzspezialisten - Diese Schlafposition solltest du unbedingt vermeiden!

 Play Caption

Additionally, meinetwegen can also suggest indifference, translated as "for all I care" or even the very rude "whatever" below. 

Machen Sie doch meinetwegen, was Sie wollen!

But do what you want, for all I care!

Caption 61, Märchen - Sagenhaft - Der Rattenfänger von Hameln

 Play Caption

„Meinetwegen“, brummte Frederick.

"Whatever," grumbled Frederick.

Caption 42, Piggeldy und Frederick - Unendlichkeit

 Play Caption

Von mir aus is a colloquial expression, and can be translated as "for all I care," "as far as I'm concerned," or even simply "I don't mind."

In welcher Form auch immer... bis morgen früh von mir aus.

In whichever form… till early morning, as far as I'm concerned.

Caption 45, Fastnacht - Karneval - Quartier Latin

 Play Caption

Von mir aus bleibe ich ewig

I don't mind remaining forever

Caption 7, Max Giesinger - Nicht so schnell

 Play Caption

Was meinst du? -Von mir aus.

What do you think? -I don't mind.

-Vielleicht hat hier jemand was gesehen.

-Maybe someone here saw something.

Caption 43, Die Pfefferkörner - Eigentor - Part 2

 Play Caption

Further Learning
Take a look for additional examples on Yabla German and try working on your own sentences with meinetwegen and responses with von mir aus. 

Not to be alarmist, but with police troubles in the USA and the rest of the troubles around the world, this might be a good time to note that although the noun "the police" is always plural in English, the noun die Polizei is dealt with quite differently in German.

BANNER PLACEHOLDER

Die Polizei ermittelt wegen Hausfriedensbruch.

The police are investigating because of [criminal] trespassing.

Caption 12, Atomkraft - Streit um AKW-Laufzeiten

 Play Caption

Ruh dich aus, bis die Polizei kommt.

Relax until the police come.

Caption 15, Die Pfefferkörner - Eigentor

 Play Caption

Die Polizei sprach von hunderttausend Menschen.

The police spoke of one hundred thousand people.

Caption 4, Papst Benedikt - Erster Rücktritt eines Papstes in der Neuzeit

 Play Caption

In English, there is no singular for the noun "the police", but in the German die Polizei, there is no plural noun form. As you can see in the above examples: die Polizei ermittelt, die Polizei kommt, and die Polizei sprach, the conjugations of the verbs used reflect a singular noun.

BANNER PLACEHOLDER

Further Learning
Do a search for the noun Polizei on Yabla German and see the many examples of how this word is used in German in a real world context!